The maximum value of Q is 16√(2/3).
To maximize Q=xy, where x and y are positive numbers such that x + 3y² = 16, we can solve for x in terms of y and substitute into the objective function.
Thus, x = 16 - 3y² and Q = (16 - 3y²)y. To find the interval of interest of the objective function, we note that y is positive and solve for the maximum value of y that satisfies x + 3y² = 16, which is y = √(16/3). Therefore, the interval of interest is (0, √(16/3)).
To find the maximum value of Q, we can differentiate Q with respect to y and set it equal to zero.
This yields 16-6y²=0, which implies y=√(16/6). Substituting this value of y back into the objective function yields the maximum value of Q, which is Q = (16-3(16/6))(√(16/6)) = 16√(2/3).
To know more about objective function click on below link:
https://brainly.com/question/29185392#
#SPJ11
what is in the middle of, 26.27 and 26.89?
The middle value between 26.27 and 26.89 is 26.58.
To find the middle value between 26.27 and 26.89, you can follow these steps:
1. Add the two numbers together: 26.27 + 26.89 = 53.16
2. Divide the sum by 2 to find the average: 53.16 ÷ 2 = 26.58
The middle value between 26.27 and 26.89 is 26.58.
The middle value of a set of numbers is commonly referred to as the "median". The median is a statistical measure of central tendency that represents the value that separates the lower and upper halves of a dataset.
To find the median of a set of numbers, the numbers must first be arranged in order from lowest to highest (or highest to lowest). If the dataset contains an odd number of values, the median is the middle value.
For example, if we have the set of numbers {1, 3, 5, 7, 9}, the median is 5, which is the value that separates the lower half {1, 3} from the upper half {7, 9}.
If the dataset contains an even number of values, the median is the average of the two middle values. For example, if we have the set of numbers {2, 4, 6, 8}, the median is (4 + 6) / 2 = 5, which is the average of the two middle values that separate the lower half {2, 4} from the upper half {6, 8}.
The median is a useful measure of central tendency because it is not affected by extreme values or outliers in the dataset, unlike the mean, which can be skewed by such values.
To know more about middle value, refer here:
https://brainly.com/question/20846131
#SPJ11
Let the plane contains the points (1,1,1),(1,2,3)&(2,1,3) parallel or perpendicular
The given points (1,1,1), (1,2,3), and (2,1,3) do not lie on a plane that is parallel or perpendicular to any given plane, since they do not satisfy the necessary conditions for either case.
To determine whether the given points lie on a plane that is parallel or perpendicular to any given plane, we need to find the normal vector of the plane containing the given points.
Let the given points be A(1,1,1), B(1,2,3), and C(2,1,3). To find the normal vector of the plane containing these points, we can take the cross product of the vectors AB and AC:
AB = <1-1, 2-1, 3-1> = <0, 1, 2>
AC = <2-1, 1-1, 3-1> = <1, 0, 2>
Normal vector N = AB x AC
= <0, 1, 2> x <1, 0, 2>
= <-2, -2, 1>
Now, to determine if the plane containing the points is parallel or perpendicular to a given plane, we need to compare the normal vector of the plane to the normal vector of the given plane. However, we are not given a plane to compare to.
Therefore, we cannot determine whether the given points lie on a plane that is parallel or perpendicular to any given plane.
Learn more about plane
https://brainly.com/question/28247880
#SPJ4
The steps for circumscribing a circle about a triangle are shown:
Step 1 Construct the angle bisector of one side of the triangle.
Step 2 Construct the perpendicular bisector of another side of the triangle.
Step 3 Indicate the point of intersection of the bisectors with a point representing the center of the circle.
Step 4 Place the compass on the center, adjust its length to reach any vertex of the triangle, and draw your circumscribed circle.
Which step is incorrect, and how can it be fixed?
a
Step 1, replace "angle bisector" with "perpendicular bisector"
b
Step 3, replace "circle" with "triangle"
c
Step 4, replace "vertex" with "side"
d
All steps are correct
There is an error in Step 1. The correct answer should be: a) Step 1, replace "angle bisector" with "perpendicular bisector"
What do the terms "inscribed" and "circumscribed" figures mean?An inscribed figure is, in essence, one shape drawn inside of another. A form drawn outside another shape is referred to as a circumscribed figure. All of a polygon's corners, or vertices, need to touch the circle in order for it to be enclosed inside of it.
The appropriate procedures for drawing a circle around a triangle are as follows:
Step 1: Create the perpendicular bisector of one triangle side .
Step 2: Create the perpendicular bisector of a different triangle side .
Step 3: Mark the place where the bisectors cross a point that represents the centre of the circle.
Step 4: Center the compass and adjust its length to reach any triangle vertex before drawing your circle's circumference.
Hence, option a is correct answer.
Learn more about circumscribed figures here:
https://brainly.com/question/1436883
#SPJ1
Determine whether the set, together with the indicated operations, is a vector space. If it is not, then identify one of the vector space axioms that fails. The set of all 4 x 4 matrices of the form [a c b 0] with the standard operations The set is a vector space. The set is not a vector space because it is not closed under addition. The set is not a vector space because an additive inverse does not exist. The set is not a vector space because it is not closed under scalar multiplication. The set is not a vector space because a scalar identity does not exist.
The set is not a vector space because it is not closed under addition.
To determine whether the given set is a vector space, we need to check if it satisfies the vector space axioms. Let's consider the closure under addition axiom. Let A = [a1 c1 b1 0] and B = [a2 c2 b2 0] be two 4x4 matrices in the set. Adding A and B, we get:
A + B = [a1+a2 c1+c2 b1+b2 0+0] = [a1+a2 c1+c2 b1+b2 0]
The resulting matrix is not guaranteed to be in the set because the sum of b1 and b2 (b1+b2) might not be equal to the sum of a1 and a2 (a1+a2). Therefore, the set is not closed under addition, and it is not a vector space.
To know more about vector space click on below link:
https://brainly.com/question/13058822#
#SPJ11
The volume of air in a person's lungs can be modeled with a periodic function. The
graph below represents the volume of air, in ml., in a person's lungs over time t,
measured in seconds.
What is the period and what does it represent in this
context?
1000
you
(2-5, 2900)
(5-5, 1100)
Time (in seconds)
(8.5, 2900)
(11.5, 1100)
The period of the function represent the given context is (8.5, 2900).
The period of this function is 8.5 seconds, and it represents the time it takes for the person's lungs to fill up with air, then empty out again.
The graph shows that the volume of air in the person's lungs is at its maximum (2900 ml) at the start of each period, and then decreases over time until it reaches its minimum (1100 ml) at the end of each period.
Therefore, the period of the function represent the given context is (8.5, 2900).
To learn more about the function visit:
https://brainly.com/question/28303908.
#SPJ1
find the normal vector to the tangent plane of z = 7 e x 2 − 4 y z=7ex2-4y at the point (8, 16, 7) component = -1.
The normal vector to the tangent plane is the opposite of this gradient, which is:
[tex]n = (-14e^{64}, 4, -1)[/tex]
What is gradient of the surface?The gradient of the surface is given by:
∇z = ( ∂z/∂x, ∂z/∂y, ∂z/∂z )
where ∂z/∂x and ∂z/∂y are the partial derivatives of z with respect to x and y, respectively.
The gradient of the surface at that point must first be determined before we can determine the normal vector to the tangent plane of the surface [tex]z=7e^{x^{2} } -4y[/tex] at the point (8, 16, 7).
Taking the partial derivatives, we get:
[tex]∂z/∂x = 14e^{x^{2} }[/tex]
∂z/∂y = -4
Plugging in the values x=8 and y=16, we get:
∂z/∂x = [tex]14e^{(8)^2} = 14e^{64}[/tex]
∂z/∂y = -4
Therefore, the gradient of the surface at the point (8, 16, 7) is:
∇z = ( [tex]14e^{64}, -4,[/tex]∂z/∂z )
The last component of the gradient (∂z/∂z) is always equal to 1, so we have:
∇z = ( [tex]14e^{64},[/tex] -4, 1 )
This gradient is perpendicular to the tangent plane of the surface at the point (8, 16, 7). Therefore, the normal vector to the tangent plane is the opposite of this gradient, which is:
n =[tex](-14e^{64}, 4, -1)[/tex]
The component of the normal vector in the z-direction is -1, as given in the problem statement.
know more about normal vector visit:
https://brainly.com/question/29050578
#SPJ1
Please answer parts a-c: (a) Sketch the graph of the function f(x) = 2*. (b) If f(x) is translated 4 units down, what is the equation of the new function g(x)? (c) Graph the transformed function g(x) on the same grid. **Both functions must be present on your graph. Remember to include at least two specific points per function! **
answer:
equation of g(x):
A graph of the function [tex]f(x) = 2^x[/tex] is shown in the image below.
If f(x) is translated 4 units down, the equation of the new function g(x) is [tex]g(x) = 2^x-4[/tex]
The transformed function g(x) is shown on the same grid below.
What is a translation?In Mathematics and Geometry, the vertical translation a geometric figure or graph downward simply means subtracting a digit from the value on the y-coordinate of the pre-image or function.
In Mathematics and Geometry, a horizontal translation to the right is modeled by this mathematical equation g(x) = f(x - N) while a vertical translation to the positive y-direction (downward) is modeled by this mathematical equation g(x) = f(x) - N.
Where:
N represents an integer.g(x) and f(x) represent functions.In this scenario, we can logically deduce that the graph of the parent function f(x) was translated or shifted downward (vertically) by 4 units as shown below.
Read more on translation here: brainly.com/question/4521517
#SPJ1
the proportion of americans with hypertension is about 33%. the claim of lbp inc is that if you take their vitamin, the chances that you get high blood pressure will go down. what are the hypotheses for this?
The null hypothesis (H0) for this probability would be that there is no effect of taking the vitamin on the chances of getting high blood pressure.
The alternative hypothesis (Ha) would be that taking the vitamin reduces the chances of getting high blood pressure.
Therefore, the hypotheses for this claim can be written as:
H0: The proportion of Americans with hypertension who take the vitamin is the same as the proportion of Americans with hypertension who do not take the vitamin.
Ha: The proportion of Americans with hypertension who take the vitamin is lower than the proportion of Americans with hypertension who do not take the vitamin.
To know more about probability,
https://brainly.com/question/30034780
#SPJ11
calculate constrained minimum find the points on the curve xy2 = 54 nearest the origin.
To calculate the constrained minimum, we need to use the method of Lagrange multipliers. We define the Lagrangian function L as L(x,y,λ) = xy^2 - λ(d^2 - x^2 - y^2 - z^2), where d represents the distance between the origin and the nearest point on the curve.
Taking the partial derivatives of L with respect to x, y, and λ, we get:
dL/dx = y^2 + 2λx = 0
dL/dy = 2xy - 2λy = 0
dL/dλ = d^2 - x^2 - y^2 - z^2 = 0
Solving these equations simultaneously, we get:
x = ± 3√6, y = ± √6, and λ = 3/2
Therefore, the points on the curve xy^2 = 54 nearest to the origin are:
(3√6, √6) and (-3√6, -√6)
These points are the constrained minimum because they are the closest points on the curve to the origin.
To find the constrained minimum and the points on the curve xy^2 = 54 nearest to the origin, we can use the method of Lagrange multipliers. Let f(x, y) = x^2 + y^2 be the distance squared from the origin, and g(x, y) = xy^2 - 54 as the constraint.
First, calculate the gradients:
∇f(x, y) = (2x, 2y)
∇g(x, y) = (y^2, 2xy)
Now, set ∇f(x, y) = λ ∇g(x, y):
(2x, 2y) = λ(y^2, 2xy)
This gives us two equations:
1) 2x = λy^2
2) 2y = λ2xy
From equation (2), we can get:
λ = 1/y
Now, substitute λ into equation (1):
2x = (1/y)y^2
2x = y
Using the constraint equation g(x, y) = xy^2 - 54 = 0, we can substitute y = 2x:
2x(2x)^2 = 54
8x^3 = 54
x^3 = 27/4
x = ∛(27/4) = 3/√4 = 3/2
Now we can find y using y = 2x:
y = 2(3/2) = 3
Thus, the point nearest to the origin on the curve xy^2 = 54 is (3/2, 3).
Visit here to learn more about Lagrangian function brainly.com/question/31367466
#SPJ11
A normally distributed population has mean 30.5 and standard deviation 3.5. Find the mean of sample mean for sample size of 10. a. 30.5 b. 0.35 c. 3.5 d. 35.0 e. 3.05
The mean of sample mean for sample size of 10 is e. 3.05.
To find the mean of sample means for a sample size of 10 from a normally distributed population with mean 30.5 and standard deviation 3.5, we use the formula:
mean of sample means = population mean = 30.5
So the answer is not affected by the sample size or standard deviation. The mean of the sample means will always be equal to the population mean.
Learn more about : Statistics - https://brainly.com/question/31497663
#SPJ11
Let A and wDrmine if w is in Col(A). Is w in Nul(A)? -2 4 Determine if w is in Col(A). Select the correct choice below and, if necessary, fill in the answer box to complete your choice. O A. The vector w is in Col(A) because the columns of A span R2 O B. The vector w is in Col(A) because Ax w is a consistent system. One solution is x ° C. The vector w is not in Col(A) because Ax = wis an inconsistent system. One row of the reduced row echelon form of the augmented matrix [A 0] has the form [0 0 b] where b The vector w is not in Col(A) because w is a linear combination of the columns of A. ( D.
B. The vector w is in Col(A) because Ax = w is a consistent system. One solution is x is the correct option.
To determine if w is in Col(A), we need to check if there exists a solution x such that Ax = w. If such a solution exists, then w is in Col(A).
However, we cannot determine if w is in Nul(A) without knowing more information about matrix A. The nullspace of a matrix is the set of all solutions x to the equation Ax = 0. It is possible for a vector to be in the column space but not in the nullspace, and vice versa.
So, the correct choice is:
B. The vector w is in Col(A) because Ax = w is a consistent system. One solution is x = (-2,4)^T.
We cannot determine if w is in Nul(A) without more information about A.
Learn more about the nullspace :
https://brainly.com/question/31323091
#SPJ11
A strip of width 4 cm is attached to one side of a square to form a rectangle. The area of the rectangle formed is 77c * m ^ 2 then find the length of the side of the square.A strip of width 4 cm is attached to one side of a square to form a rectangle. The area of the rectangle formed is 77c * m ^ 2 then find the length of the side of the square.
Answer:
x = -2 + sqrt(4 + 77c*m^2) cm.
Step-by-step explanation:
Let's denote the side of the square by "x".
When the strip of width 4 cm is attached to one side of the square, the resulting rectangle has dimensions of (x+4) cm by x cm.
The area of the rectangle is given by:
(x+4) * x = 77c * m^2
Expanding the left-hand side and simplifying, we get:
x^2 + 4x = 77c * m^2
Moving all the terms to one side, we get:
x^2 + 4x - 77c * m^2 = 0
Now we can use the quadratic formula to solve for x:
x = [-4 ± sqrt(4^2 - 41(-77cm^2))] / (21)
x = [-4 ± sqrt(16 + 308cm^2)] / 2
x = [-4 ± 2sqrt(4 + 77cm^2)] / 2
x = -2 ± sqrt(4 + 77c*m^2)
Since the length of a side of a square must be positive, we can discard the negative solution and get:
x = -2 + sqrt(4 + 77c*m^2)
Therefore, the length of the side of the square is x = -2 + sqrt(4 + 77c*m^2) cm.
The length of the side of the square is x = -2 + 2√(1 + 19c) cm.
What is Area of Rectangle?The area of Rectangle is length times of width.
Let the side of the square be x cm.
When a strip of width 4 cm is attached to one side of the square, the resulting rectangle will have dimensions (x + 4) cm and x cm.
The area of the rectangle is given as 77c m² so we have:
(x + 4)x = 77c
Expanding the left side, we get:
x² + 4x = 77c
Bringing all the terms to one side, we have:
x² + 4x - 77c = 0
Now, we can use the quadratic formula to solve for x:
x = [-4 ± √(4² - 4(1)(-77c))] / 2(1)
x = [-4 ± √(16 + 308c)] / 2
x = [-4 ± √(16(1 + 19c))] / 2
x = [-4 ± 4√(1 + 19c)] / 2
x = -2 ± 2√(1 + 19c)
Since x must be positive, we take the positive root:
x = -2 + 2√(1 + 19c)
Therefore, the length of the side of the square is x = -2 + 2√(1 + 19c) cm.
To learn more on Area of Rectangle click:
https://brainly.com/question/20693059
#SPJ2
Consider the sum 10 + 21 +32 +43 + ... +406. A. How many terms (summands) are in the sum? B. Compute the sum using a technique discussed in this section.
A. There are 37 terms in the sum.
B. The sum of the given series is 7,696.
How many terms are in the sum?A. Using arithmetic sequences, We can observe that each term in the sum is obtained by adding 11 to the previous term. Therefore, the nth term can be expressed as:
[tex]a_n = 10 + 11(n-1)[/tex]
We want to find the number of terms in the sum up to [tex]a_n[/tex] = 406. Setting [tex]a_n[/tex]= 406 and solving for n, we get:
406 = 10 + 11(n-1)
396 = 11(n-1)
n = 37
Therefore, there are 37 terms in the sum.
How to compute the sum?B. We can use the formula for the sum of an arithmetic series:
[tex]S_n = n/2 * (a_1 + a_n)[/tex]
where [tex]S_n[/tex] is the sum of the first n terms, [tex]a_1[/tex] is the first term, and[tex]a_n[/tex] is the nth term.
In this case, we have:
[tex]a_1[/tex]= 10
[tex]a_n[/tex]= 406
n = 37
Substituting these values, we get:
[tex]S_{37}[/tex] = 37/2 * (10 + 406)
[tex]S_{37}[/tex] = 18.5 * 416
[tex]S_{37}[/tex] = 7,696
Therefore, the sum of the given series is 7,696.
Learn more about arithmetic sequences
brainly.com/question/15412619
#SPJ11
..if there were 10 customers and your expenses are about 52 dollars, how much is your profit and revenue?
Answer:
Step-by-step explanation:
The table shows the number of students who signed up for different after school activities. Each student signed up for exactly one activity.
Activity Students
Cooking
9
99
Chess
4
44
Photography
8
88
Robotics
11
1111
Total
32
3232
Match the following ratios to what they describe.
Description
Ratio
a) The ratio of the photography students to the chess students is 2 : 1
b) The ratio of photography students to all students is 1 : 4
c) The ratio of chess students to all students is 1 : 8
Given data ,
Let the number of cooking students = 9
Let the number of chess students = 4
Let the number of photography students be = 8
Let the number of robotics students = 11
So , the total number of students = 32
a)
The ratio of the photography students to the chess students = 8 / 4
On simplifying the proportion , we get
The ratio of the photography students to the chess students = 2 : 1
b)
The ratio of photography students to all students = 8 / 32
The ratio of photography students to all students = 1 : 4
c)
The ratio of chess students to all students = 4 / 32
The ratio of chess students to all students = 1 : 8
Hence , the proportion is solved
To learn more about proportion click :
https://brainly.com/question/7096655
#SPJ1
The complete question is attached below :
The table shows the number of students who signed up for different after school activities. Each student signed up for exactly one activity.
HURRY UP Please answer this question
Answer:
b=32
Step-by-step explanation:
a^2+b^2=c^2
24^2+b^2=40^2
576+b^2=1600
b^2=1600-576
b^2=1024
b=32
what is equivalent to 5x - 19
For x = 3, the expression 5x - 19 is equivalent to -4.
Here, we have an expression, 5x - 19, which is not an equation yet because it doesn't have an equal sign (=). An equation requires that both sides of the equal sign be equivalent.
To make this expression into an equation, we need to set it equal to something. Let's say we want to find an expression that is equivalent to 5x - 19. We can represent this unknown expression as "y". So, our equation will be:
5x - 19 = y
This equation is now saying that the expression 5x - 19 is equivalent to y. We can also say that "y" is a function of "x", meaning that the value of "y" depends on the value of "x".
We can use this equation to find the value of "y" for any given value of "x". If we substitute x = 3, we get:
5(3) - 19 = y
y = 15 - 19
y = -4
To know more about equation here
https://brainly.com/question/10413253
#SPJ1
Complete Question:
what is equivalent to 5x - 19 when x = 3?
let {n,k} denote the number of partitions of n distinct objects into k nonempty subsets. show that {n+1,k}=k{n,k}+{n,k-1}
The total number of ways to partition the set of n+1 distinct objects into k nonempty subsets is {n+1,k} = k{n,k} + {n,k-1}, as required.
To show that {n+1,k}=k{n,k}+{n,k-1}, we can use a combinatorial argument.
Consider a set of n+1 distinct objects. We want to partition this set into k nonempty subsets. We can do this in two ways
Choose one of the n+1 objects to be the "special" object. Then partition the remaining n objects into k-1 nonempty subsets. This can be done in {n,k-1} ways.
Partition the n+1 objects into k nonempty subsets, and then choose one of the subsets to be the subset that contains the special object. There are k ways to choose the subset that contains the special object, and once we have chosen it, we need to partition the remaining n objects into k-1 nonempty subsets. This can be done in {n,k-1} ways.
Learn more about combinatorial argument here
brainly.com/question/28452452
#SPJ4
Evaluate the iterated integral by converting to polar coordinates.
4
∫ ^√16 − x2 e^−x^2 − y^2 dy dx
0
____
The final answer will be an approximation.
We have the iterated integral:
∫(0 to √16) ∫(0 to √[tex]16-x^2) e^(-x^2-y^2) dy dx[/tex]
To convert to polar coordinates, we use the substitution x = r cos(θ) and y = r sin(θ), where r is the radial distance and θ is the angle with the positive x-axis.
The limits of integration also need to be changed accordingly. For the inner integral, we have:
0 ≤ y ≤ √[tex](16-x^2)[/tex]
Substituting y = r sin(θ), we get:
0 ≤ r sin(θ) ≤ √(16 - [tex]r^2 cos^2[/tex](θ))
Squaring both sides, we get:
0 ≤ [tex]r^2 sin^2[/tex](θ) ≤ 16 - [tex]r^2 cos^2[/tex](θ)
Rearranging, we get:
[tex]r^2 (cos^2[/tex](θ) + [tex]sin^2[/tex](θ)) ≤ 16
[tex]r^2[/tex] ≤ 16
0 ≤ r ≤ 4
For the outer integral, we have:
0 ≤ x ≤ √16
Substituting x = r cos(θ), we get:
0 ≤ r cos(θ) ≤ √16
0 ≤ r ≤ 4 cos(θ)
Thus, the integral in polar coordinates becomes:
∫(0 to π/2) ∫(0 to 4 cos(θ)) [tex]e^(-r^2[/tex]) r dr dθ
Evaluating the inner integral, we get:
∫(0 to 4 cos(θ)) e^(-r^2) r dr = -1/2 e^(-r^2) |0 to 4 cos(θ)) = 1/2 (1 - e^(-16 cos^2(θ)))
Substituting back into the outer integral, we get:
∫(0 to π/2) 1/2 (1 - e^(-16 cos^2(θ))) dθ
Simplifying, we get:
1/2 ∫(0 to π/2) (1 - e^(-16 cos^2(θ))) dθ
To evaluate this integral, we need to use numerical methods or approximations. Therefore, the final answer will be an approximation.
To learn more about approximation. visit:
https://brainly.com/question/30707441
#SPJ11
calculate the euclidean distance between the following two points: (5,8,-2) and (-4,5,3) round to two decimal places
The Euclidean distance between the two points (5,8,-2) and (-4,5,3) is approximately 10.72 (rounded to two decimal places).
Explanation: -
To calculate the Euclidean distance between two points (5,8,-2) and (-4,5,3), you can use the following formula:
Euclidean Distance = √((x2 - x1)^2 + (y2 - y1)^2 + (z2 - z1)^2)
Here, (x1, y1, z1) = (5, 8, -2) and (x2, y2, z2) = (-4, 5, 3). Now, plug in the values:
Step 1. Calculate the differences between the co-ordinates;
(x2 - x1) = (-4 - 5) = -9
(y2 - y1) = (5 - 8) = -3
(z2 - z1) = (3 - (-2)) = 5
Step 2. Square the differences:
(-9)^2 = 81
(-3)^2 = 9
(5)^2 = 25
Step 3. Add the squared differences:
81 + 9 + 25 = 115
Step 4. Take the square root:
√115 ≈ 10.72
So, the Euclidean distance between the two points is approximately 10.72 (rounded to two decimal places).
Know more about the "Euclidean distance" click here:
https://brainly.com/question/30598325
#SPJ11
] a bacteria culture starts with e 10 bacteria and grows at a rate proportional to its size. after 3 hours there are e 16 bacteria. 1. find an expression for the number of bacteria after t hours.
An expression for the number of bacteria after t hours is [tex]N(t) = e^{(10 + 2t)}[/tex].
To find an expression for the number of bacteria after t hours, given that a bacteria culture starts with e¹⁰ bacteria and grows at a rate proportional to its size, we can use the exponential growth formula:
[tex]N(t) = N(0) * e^{(kt)}[/tex]
where N(t) is the number of bacteria after t hours, N(0) is the initial number of bacteria, k is the growth constant, and t is the time in hours.
First, we are given that after 3 hours there are e¹⁶ bacteria.
Let's use this information to find the growth constant k:
[tex]e^{16} = e^{10} * e^{(3k)} \\e^{6} = e^{(3k)}[/tex]
Now, we can find k by dividing both sides by 3:
6 = 3k
k = 2
Now that we have the growth constant k, we can plug it into the exponential growth formula:
[tex]N(t) = e^{10} * e^{(2t)}[/tex]
which is equivalent to:
[tex]N(t) = e^{(10 + 2t)}[/tex]
Learn more about expression:
https://brainly.com/question/723406
#SPJ11
Need help asap! thanks!
Yes the opposite sides of the figure are congruent because:
Both WX and YZ have a length of 3.6
Both XY and WZ have a length of 16.8
How to Use Pythagoras Theorem?Pythagoras Theorem is the way in which you can find the missing length of a right angled triangle.
The triangle has three sides, the hypotenuse (which is always the longest), Opposite (which doesn't touch the hypotenuse) and the adjacent (which is between the opposite and the hypotenuse).
Pythagoras Theorem is in the form of;
a² + b² = c²
Using Pythagoras theorem, we have:
WX = √(3² + 2²)
WX = √13 = 3.6
YZ = √(3² + 2²)
YZ = √13 = 3.6
Similarly:
XY = √(5² + 16²)
XY = √281
XY = 16.8
WZ = √(5² + 16²)
WZ = √281
WZ = 16.8
Read more about Pythagoras Theorem at: https://brainly.com/question/231802
#SPJ1
Yes the opposite sides of the figure are congruent because:
Both WX and YZ have a length of 3.6
Both XY and WZ have a length of 16.8
How to Use Pythagoras Theorem?Pythagoras Theorem is the way in which you can find the missing length of a right angled triangle.
The triangle has three sides, the hypotenuse (which is always the longest), Opposite (which doesn't touch the hypotenuse) and the adjacent (which is between the opposite and the hypotenuse).
Pythagoras Theorem is in the form of;
a² + b² = c²
Using Pythagoras theorem, we have:
WX = √(3² + 2²)
WX = √13 = 3.6
YZ = √(3² + 2²)
YZ = √13 = 3.6
Similarly:
XY = √(5² + 16²)
XY = √281
XY = 16.8
WZ = √(5² + 16²)
WZ = √281
WZ = 16.8
Read more about Pythagoras Theorem at: https://brainly.com/question/231802
#SPJ1
suppose p is a prime number and p2 divides ab and gcd(a,b)=1. Show p2 divides a or p2 divides b.
p is a prime number
p^2 divides ab with gcd(a, b) = 1,
then p^2 divides a or p^2 divides b.
Fundamental Theorem of Arithmetic:
1. Since gcd(a, b) = 1, we know that a and b are coprime, meaning they have no common factors other than 1.
2. Given that p is a prime number and p^2 divides ab, this implies that p divides either a or b (or both) due to the Fundamental Theorem of Arithmetic.
3. Let's assume p divides a. Then, we can write a = pk for some integer k.
4. Now, we know that p^2 divides ab, which means ab = p^2m for some integer m.
Substitute a with pk from step 3: ab = (pk)b.
5. Thus, p^2m = pkb. Since p is a prime number, by Euclid's Lemma, we know that p must divide either kb or b itself. We already assumed p divides a, so p cannot divide b (as gcd(a, b) = 1). Therefore, p must divide kb.
6. As p divides a (a = pk) and p divides kb, we can conclude that p^2 divides a. So, p^2 divides a.
7. If we instead assumed p divides b, we would arrive at a similar conclusion: p^2 divides b.
In summary, if p is a prime number and p^2 divides ab with gcd(a, b) = 1, then either p^2 divides a or p^2 divides b.
To know more about Fundamental Theorem of Arithmetic:
https://brainly.com/question/15317294
#SPJ11
The first four nonzero terms in the power series expansion of the function f(x) = sinx about x = 0 are Select the correct answer. Oa 1-x+x2/2-x3 /3 Ob.1-x2/2+x4/24-x6/720 Odx-x3/6+x/120-x/5040 Oe. 1 +x2 / 2 +x4 / 4 +x" / 6
The first four nonzero terms in the power series expansion option (D) - [tex]x^{3/6} + x^{5/120} - x^{7/5040[/tex]
What is function?In mathematics, a function is a unique arrangement of the inputs (also referred to as the domain) and their outputs (sometimes referred to as the co-domain), where each input has exactly one output and the output can be linked to its input.
The power series expansion of the function f(x) = sin(x) about x = 0 is given by:
f(x) = [tex]x - x^{3/3}! + x^{5/5}! - x^{7/7}! + ...[/tex]
To find the first four nonzero terms, we substitute x = 0 into the series and discard all the terms that are zero:
f(0) = 0 - 0 + 0 - 0 + ... = 0
f'(0) = 1 - 0 + 0 - 0 + ... = 1
f''(0) = 0 - 3!/2! + 0 + 0 + ... = -3/2
f'''(0) = 0 + 0 + 5!/3! - 0 + ... = 5/6
Therefore, the first four nonzero terms in the power series expansion of sin(x) about x = 0 are:
sin(x) ≈ [tex]x - x^{3/3}! + x^{5/5}! - x^{7/7}![/tex]
Simplifying this expression, we get:
sin(x) ≈ [tex]x - x^{3/6} + x^{5/120} - x^{7/5040[/tex]
So, the correct answer is option (D) - [tex]x^{3/6} + x^{5/120} - x^{7/5040[/tex].
Learn more about function on:
https://brainly.com/question/10439235
#SPJ11
PLS HELP VERY CONFUSED!! AABC has vertices at (-4, 4), (0,0) and (-5,-2). Find the coordinates of points A, B and C after a reflection across y = -x.
Answer:
A' = (-4, 4)
B' = (0, 0)
C' = (2, 5)
Step-by-step explanation:
When a point is reflected across the line y = -x, the x-coordinate becomes -y, and the y-coordinate becomes -x. Therefore, the mapping rule is:
(x, y) → (-y, -x)Given vertices of triangle ABC:
A = (-4, 4)B = (0, 0)C = (-5, -2)Therefore, if we reflect the given points across the line y = -x, the coordinates of the reflected points are:
[tex]\begin{aligned}& \sf A = (-4, 4)& \implies\;\; \sf A'& =\sf (-4,-(-4))=(-4,4)\\& \sf B = (0, 0) &\implies\;\; \sf B' &= \sf (-0, -0)=(0,0)\\& \sf C = (-5, -2)& \implies\;\; \sf C' &= \sf (-(-2),-(-5))=(2,5)\end{aligned}[/tex]
Pls help me asap!!!
On the axes below, make an appropriate scale and graph exactly one cycle of the trigonometric function y = 7 sin 6x.
The graph is given in the image below:
How to make the right scale for the trig functionTo plot a full cycle of y = 7sin(6x), begin by dividing the period (2π) by six to obtain π/3, which is then used to mark every increment of π/6 along the x-axis.
Additionally, since y ranges from -7 to 7, label the y-axis in increments of either 1 or 2.
Plot the key points at (0,0), (π/12,7), (π/6,0), (π/4,-7), and (π/3,0), and finally connect them smoothly with a curve to complete the plot of one full cycle.
Read more about trigonometric functions here:
https://brainly.com/question/25618616
#SPJ1
Answer this math question for 15 points
Answer:
(the length of leg)^2=87^2-60^2=3969
so when take the root :
the length of leg=63ft
Find f.
f '(x) = √x(3+10x)
f (1) = 9
f (x) = ____
The function f (x) = 2x^(3/2) + (4/3)x^(5/2) + (17/3)
To find the function f(x), given that f'(x) = √x(3+10x) and f(1) = 9, follow these steps:
1. Integrate f'(x) with respect to x to find f(x).
∫(√x(3+10x)) dx
2. Perform a substitution to make the integration easier. Let u = x, then du = dx.
∫(u^(1/2)(3+10u)) du
3. Now, distribute the u^(1/2) term and integrate term by term:
∫(3u^(1/2) + 10u^(3/2)) du
4. Integrate each term:
[2u^(3/2) + (4/3)u^(5/2)] + C
5. Replace u with x:
f(x) = [2x^(3/2) + (4/3)x^(5/2)] + C
6. Use the given point f(1) = 9 to find the value of the constant C:
9 = [2(1)^(3/2) + (4/3)(1)^(5/2)] + C
9 = 2 + (4/3) + C
C = 9 - 2 - (4/3)
C = 7 - (4/3)
C = (17/3)
7. Plug the value of C back into f(x):
f(x) = [2x^(3/2) + (4/3)x^(5/2)] + (17/3)
So, the function f(x) is given by:
f(x) = 2x^(3/2) + (4/3)x^(5/2) + (17/3)
Know more about function here:
https://brainly.com/question/11624077
#SPJ11
explain why the set of natural numbers {1,2,3,4,...} and the set of even numbers {2, 4, 6, 8, . . .} have the same cardinality.
The sets of natural numbers {1, 2, 3, 4, ...} and even numbers {2, 4, 6, 8, ...} have the same cardinality because there exists a bijective function between the two sets. A bijective function is a one-to-one correspondence that pairs each element in one set with exactly one element in the other set. In this case, the function f(n) = 2n pairs each natural number n with an even number 2n, ensuring that the two sets have the same cardinality.
The two sets, the set of natural numbers {1,2,3,4,...} and the set of even numbers {2, 4, 6, 8, . . .}, have the same cardinality because we can create a one-to-one correspondence between the two sets. To do this, we can simply map each natural number to its corresponding even number (i.e., 1 maps to 2, 2 maps to 4, 3 maps to 6, and so on). This mapping covers all elements of both sets, without skipping any, and without duplicating any. Thus, the two sets have the same number of elements, which means they have the same cardinality.
Learn more about natural numbers here: brainly.com/question/17429689
#SPJ11
TEXT ANSWER
2x + 5y = -10
-2x + y = 46
SHOW ALL WORK in the way that works best for you
Answer:
x= - 20
y= 6
Step-by-step explanation:
2x + 5y = -10
-2x + y = 46
if u solve it
it is 6y= 36 so y =6
and substitute y value into the equation:
-2x+ 6 = 46
-2x= 40
so x= -20