Solve the following two problems using Calculus. Show all of your work on a separate sheet of paper, making clear how you arrived at all your solutions. Please be aware that you are using similar thinking for both of these problems, but you are solving for maximum area in Part A and minimum cost in Part B. Enjoy

A. If the materials for the fence cost $12 per foot, find the dimensions of the
corral for the largest possible area that can be enclosed with $2400 worth of fence.

B. If he only requires an area of 288 square feet for his vegetable garden, find the minimum cost of putting up this fence, if the cost per foot is $10. Indicate the dimensions of the garden that will minimize the cost.​

The functions are illustrations of trigonometry ratios

[tex]\mathbf{(a)\ cos\theta = \frac{\sqrt{15}}{5}}[/tex]

The sine of an angle is calculated as follows

[tex]\mathbf{sin\theta = \sqrt{1 - cos^2\theta}}[/tex]

So, we have:

[tex]\mathbf{sin\theta = \sqrt{1 - (\sqrt{15}/5)}}[/tex]

[tex]\mathbf{sin\theta = \sqrt{1 - (15/25)}}[/tex]

Take LCM

[tex]\mathbf{sin\theta = \sqrt{\frac{10}{25}}}[/tex]

[tex]\mathbf{sin\theta = \frac{\sqrt{10}}{5}}[/tex]

[tex]\mathbf{(b)\ tan\theta = \frac{\sqrt{10}}{2}}[/tex]

The tangent of an angle is:

[tex]\mathbf{\ tan\theta = \frac{Opposite}{Adjacent}}[/tex]

Using Pythagoras theorem, we have:

[tex]\mathbf{Hypotenuse^2 = Opposite^2 + Adjacent^2}[/tex]

So, we have:

[tex]\mathbf{Hypotenuse^2 = (\sqrt{10})^2 + 2^2}[/tex]

[tex]\mathbf{Hypotenuse^2 = 10 + 4}[/tex]

[tex]\mathbf{Hypotenuse^2 = 14}[/tex]

Take square roots

[tex]\mathbf{Hypotenuse = \sqrt{14}}[/tex]

So, the sine of the angle is:

[tex]\mathbf{\ sin\theta = \frac{Opposite}{Hypotenuse}}[/tex]

[tex]\mathbf{\ sin\theta = \frac{\sqrt{10}}{\sqrt{14}}}[/tex]

Rationalize

[tex]\mathbf{\ sin\theta = \frac{\sqrt{140}}{14}}[/tex]

[tex]\mathbf{(c)\ csc\theta = \frac{\sqrt{15}}{3}}[/tex]

The sine of the angles is:

[tex]\mathbf{sin \theta = \frac{1}{csc\theta}}[/tex]

So, we have:

[tex]\mathbf{sin \theta = \frac{1}{\sqrt{15}/3}}[/tex]

[tex]\mathbf{sin \theta = \frac{3}{\sqrt{15}}}[/tex]

Rationalize

[tex]\mathbf{sin \theta = \frac{3\sqrt{15}}{15}}[/tex]

[tex]\mathbf{sin \theta = \frac{\sqrt{15}}{5}}[/tex]

[tex]\mathbf{(d)\ sec\theta =\frac{\sqrt 6}{3}}}[/tex]

The sine of an angle is calculated as follows

[tex]\mathbf{sin\theta = \sqrt{1 - (1/sec\theta)^2}}[/tex]

So, we have:

[tex]\mathbf{sin\theta = \sqrt{1 - (3/\sqrt 6)^2}}[/tex]

[tex]\mathbf{sin\theta = \sqrt{(1 - 9/6)}}[/tex]

[tex]\mathbf{sin\theta = \sqrt{( - 3/6)}}[/tex]

The sine of the angle does not have a real value

[tex]\mathbf{(e)\ cot\theta = \frac{\sqrt 6}{2}}[/tex]

Start by calculating the tangent of the angle

[tex]\mathbf{tan\theta = \frac{1}{cot\theta}}[/tex]

So, we have:

[tex]\mathbf{tan\theta = \frac{1}{\sqrt 6/2}}[/tex]

[tex]\mathbf{tan\theta = \frac{2}{\sqrt 6}}[/tex]

Rationalize

[tex]\mathbf{tan\theta = \frac{2\sqrt 6}{6}}[/tex]

[tex]\mathbf{tan\theta = \frac{\sqrt 6}{3}}[/tex]

The tangent of an angle is:

[tex]\mathbf{\ tan\theta = \frac{Opposite}{Adjacent}}[/tex]

Using Pythagoras theorem, we have:

[tex]\mathbf{Hypotenuse^2 = Opposite^2 + Adjacent^2}[/tex]

So, we have:

[tex]\mathbf{Hypotenuse^2 = (\sqrt{6})^2 + 3^2}[/tex]

[tex]\mathbf{Hypotenuse^2 = 6 + 9}[/tex]

[tex]\mathbf{Hypotenuse^2 = 15}[/tex]

Take square roots

[tex]\mathbf{Hypotenuse = \sqrt{15}}[/tex]

So, the sine of the angle is:

[tex]\mathbf{\ sin\theta = \frac{Opposite}{Hypotenuse}}[/tex]

[tex]\mathbf{\ sin\theta = \frac{\sqrt{6}}{\sqrt{15}}}[/tex]

Rationalize

[tex]\mathbf{\ sin\theta = \frac{\sqrt{90}}{15}}[/tex]

Rewrite as:

[tex]\mathbf{\ sin\theta = \frac{3\sqrt{10}}{15}}[/tex]

[tex]\mathbf{\ sin\theta = \frac{\sqrt{10}}{5}}[/tex]

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Answer:

7/12

1/6 to 2/12 add 5/12 to get 7/12

Answer:

0884

Step-by-step explanation:

Answer:

A is the answer

trust me

here is your answer and please follow me and Mark me As brain list

A manufacturer of bicycles has determined that 550 bikes per month will be sold at a price of $250. At a price of $150, it is estimated that 750 bikes will be sold.

a) Determine a linear function (equation) that predicts the number of bikes that will be sold each month at any given price.

b) Use your equation from question a to predict how many bikes will be sold at $350.

Step-by-step explanation:

Option A os Correct

Answer:

niether do i

Step-by-step explanation:

Answer:

okay, well, is there any link u wanna post for us to help u? maybe we can explain it?

Step-by-step explanation:

Answer:

450,000

Step-by-step explanation:

The ten thousand digit is 4 and if we compare it to 9, it is bigger then 5 therefore it is rounded up and you get the answer of 450,000

Answer: he is currently 69 years old

Step-by-step explanation:

24x3=72-3=69 lol funni number

Answer:

78 years old

Step-by-step explanation:

In 3 years Joanne will be 27. Xavier is 3 times that age by the time Joanne is 27. 27×3=81. but we're looking for Xaviers age today. so we take 81 and subtract 3 which leaves us with 78

Answer:

sorry I can't understand the word

The constraints in the question are given by the definition of the variables;

The constraints are represented by the following equation and inequalities.

Reasons:

The budget of the principal = $225

Number of people expected = 16 people

Number of sandwich per person = 2

Cost of each sandwich = $3

Required:

To select the inequalities that represent the situation's constraint

Solution:

The variable that represent the number of people attending = n

The number of sandwiches = s

From the question , we have;

The number of sandwich per attendee = 2 sandwiches

The total number of sandwich, s, for n attendees = 2·n

Therefore, we have;

s = 2·n

The number of people attending is at least 16, therefore;

n ≥ 16

The cost of the s sandwich = $3 × s

The total cost should be less than or equal to given budget of $225,

therefore;

3s ≤ 225

The equations and inequalities that could represent the constraints are;

A. n ≥ 16, D. s = 2·n, and F. 3s ≤ 225

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Answer: =11/4

Step-by-step explanation:

Answer:

[tex]y = 2 \frac{3}{4} \\ [/tex]

Step-by-step explanation:

[tex]5(y - 1) = 3(2y - 5) - (1 - 3y) \\ 5y - 5 = 6y - 15 - 1 + 3y \\ 5y - 5 = 9y - 16 \\ 16 - 5 = 9y - 5y \\ 11 = 4y \\ \frac{11}{4} = y \\ 2 \frac{3}{4} = y \\ [/tex]

Answer:

What makes a number friendly?

In number theory, friendly numbers are two or more natural numbers with a common abundancy index, the ratio between the sum of divisors of a number and the number itself. Two numbers with the same "abundancy" form a friendly pair; n numbers with the same "abundancy" form a friendly n-tuple.

Step-by-step explanation:

Answer:

y = -3x + 6

Step-by-step explanation:

Two lines that are parallel will have the same slope. All we need to do is find the new y-intercept.

-3 = -3(3) + b

-3 = -9 + b

6 = b

Now we can input the stuff we know and solved for back into the original formula [ y = mx + b ].

y = -3x + 6

Best of Luck!

A parallel line means our new line will have a slope equal to the original line.

The original line is in slope-intercept form, y = mx + b, where m is the slope and b is the value of the y-intercept. This tells us that the slope we want to use is m = -3

By using a slope of -3, with the given point (3,-4), we can now use the point-slope formula to find our new line.

Point-slope formula: y - y1 = m(x - x1) where m is the slope and the line passes through the point (x1,y1)

Combining our point and slope, we get:

y + 4 = -3(x - 3)

If we wish, we can rearrange this by distributing on the right hand side.

y + 4 = -3x + 9

We can then subtract 4 from both sides to get our answer in slope-intercept form

y + 4 - 4 = -3x + 9 - 4

y = -3x + 5

Answer: A  F D G

Step-by-step explanation:

Answer:

C) Minimum at (3,7)

Step-by-step explanation:

Recall that vertex form is [tex]y=a(x-h)^2+k[/tex]:

[tex]y=x^2-6x+16[/tex]

[tex]y=(x^2-6x+9)+7[/tex]

[tex]y=(x-3)^2+7[/tex]

Therefore, the vertex is [tex](h,k)[/tex] or [tex](3,7)[/tex] in this case. Because the parabola is positive, then the vertex has to be the minimum of the parabola, making choice C correct.

Answer:

Step-by-step explanation:

4x2 - 8x -12 = 0

divide through by 4

x2 - 2x - 3 = 0

x2 - 3x + x - 3 = 0

(x2 - 3x) + (x - 3) = 0

x(x - 3) + 1(x - 3) = 0

(x + 1) (x - 3) = 0

x + 1 = 0 and x - 3 = 0

x = -1 and x = 3

Answer:

B IS THE ANSWER

Step-by-step explanat

(22x2 -  8x) -  12  = 0

  4x2 - 8x - 12  =   4 • (x2 - 2x - 3)

The first term is,  x2  its coefficient is  1 .

The middle term is,  -2x  its coefficient is  -2 .

The last term, "the constant", is  -3  

Step-1 : Multiply the coefficient of the first term by the constant   1 • -3 = -3  

Step-2 : Find two factors of  -3  whose sum equals the coefficient of the middle term, which is   -2 .

     -3    +    1    =    -2    That's it

Step-3 : Rewrite the polynomial splitting the middle term using the two factors found in step 2 above,  -3  and  1  

                    x2 - 3x + 1x - 3

Step-4 : Add up the first 2 terms, pulling out like factors :

                   x • (x-3)

             Add up the last 2 terms, pulling out common factors :

                    1 • (x-3)

Step-5 : Add up the four terms of step 4 :

                   (x+1)  •  (x-3)

            Which is the desired factorization

4 • (x + 1) • (x - 3)  = 0

A product of several terms equals zero.

When a product of two or more terms equals zero, then at least one of the terms must be zero.

We shall now solve each term = 0 separately  

Any solution of term = 0 solves product = 0 as well.

  Solve  :    x+1 = 0

Subtract  1  from both sides of the equation :  

                     x = -1

   Solve  :    x-3 = 0  

Add  3  to both sides of the equation :  

                     x = 3

Price of one unit Sausage =  1.65

Sum ( Price x Quantity) = Total Expenditure  

Price of Bread x Quantity of Bread + Price of Ketchup x Quantity of Ketchup + Price of Sausage x Quantity of Sausage

Total Expenditure Value = 10 - 0.45  = 9.55

Let price of sausage = P

Therefore, 2 (1.40) + 1 (1.80) + 3 P = 9.55

2.80 + 1.80 + 3 Ps = 9.55

4.6 + 3 P = 9.55

3P = 9.55 - 4.60 =  4.95

P = 4.95 / 3 = 1.65

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Answer:

f(8) = 13

Step-by-step explanation:

Step 1: Plug in 8 for the value of x.

Step 2: Multiply 2 by 8.

Step 3: Subtract 3 from 16.

Therefore, f(8) = 13.

Answer:

Step-by-step explanation:

Hi there!

Given;

f(x) = 2x - 3

To find: f(8)

Replace 8 in each place where there is "x" in f(x).

or, f(8) = 2*8 -3

           = 16-3

           = 13

Therefore, f(8) = 13.

Hope it helps!

Answer:

6x - 14

Step-by-step explanation:

4(2x - 3) - 2(x + 1)

1. Expand both brackets

(8x - 12) - (2x + 2)

2. Simplify

8x - 2x = 6x

-12 - 2 = -14

Therefore, 4(2x - 3) - 2(x + 1) = 6x - 14

Step-by-step explanation:

first you need to tranfer 1 to RHS do the addition

n/3 = 8+1

=) n/3 = 9

then you need to transfer 3 to RHS and multiply

=> n = 9 x 3

there for n = 27

Answer:

1. Triangle XYZ undergoes a translation.

2. Triangle X'Y'Z' undergoes a reflection across the x-axis

Answer: 5 to 20 intervals.

Step-by-step explanation:

Answer:

-3.392

Step-by-step explanation:

Math-way

i had to do the answer as a pic cuz my iPad is gonna die.

sorry <:)

Answer:

Step-by-step explanation:

x = - 23

y = - 5

z = - 10

(x - 12)/(y+z)=

(-23 - 12)/(-5-10)=

45/15=

7/3

Answer:32.5

Step-by-step explanation: i used a calculator

Answer:

$162.50

Step-by-step explanation:

Given :   Isaac is purchasing two pairs of shoes—one pair for $37.00 and the second pair for $42.00. The state sales tax applied to Isaac’s bill is 7%.

To Find : How much is Isaac’s total bill?  

b) 25% discount off  before tax

Solution:

one pair for $37.00

second pair for $42.00

Total Bill before taxes = 37 + 42 =  79 $

7 % Sale Tax = (7/100)79 = 5.53 $

Isaac’s total bill = 79 +  5.53  = 84.53  $

Total Bill before taxes = 37 + 42 =  79 $

Discount 25 % = (25/100)79 = 19.75$

Total Bill before taxes = 79 - 19.75 = 59.25 $

7 % Sale Tax = (7/100)59.25 =4.1475 $

Isaac’s total bill = 59.25  +  4.1475  = 63.3975  $

Please post Question one by one

Learn More:

The sales tax on a surfboard is $12. What is the percent of sales tax ...

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An article is avilable for rs1540 inclusive of sales tax at 10% .what ...

Answer:

the answer is 5.50 or the last one

Step-by-step explanation:

hope this helps

Answer:

answer is $7.5

Step-by-step explanation:

1shirt=$5

10 shirt= ?

10*5= $50

$5= 100%

what about 150%

(5*150)÷100= $7.5