I can't figure this problem out- Please help me​

Printer A has the better rate of cost per page because the approximate rate of Printer A, $0.24 per page, is greater than the approximate rate of Printer B, $0.23 per page.

Rate demonstrates how many times one number can fit into another number. It contrast two numbers by ordinarily dividing them. A/B will be the formula if one is comparing one data point (A) to another data point (B).

In this case, Printer A prints 100 pages for $23.99. The rate is:

[tex]\sf = \dfrac{\$23.99}{100}[/tex]

[tex]\sf = 0.2399\thickapprox\$0.24 \ per \ page[/tex]

Printer B prints 275 sheets for $63.99. The rate is:

[tex]\sf = \dfrac{\$63.99}{275}[/tex]

[tex]\sf = 0.2327\thickapprox\$0.23 \ per \ page[/tex]

Therefore, based on the above calculations, we can see that Printer A has the better rate of cost per page because the approximate rate of Printer A, $0.24 per page, is greater than the approximate rate of Printer B, $0.23 per page.

So option (A) is correct.

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The slant height of the cone is approximately 29.29 cm to the nearest tenth of a centimeter.

To solve the problem, we must first understand what the lateral area of a cone is. The lateral area of a cone is the curved surface area of the cone, which does not include the area of the base. It can be calculated by multiplying the slant height of the cone by the circumference of the base.

Using the formula for the lateral area of a cone, we can write:

Lateral area = πrℓ

where r is the radius of the base, and ℓ is the slant height.

Substituting the given values of lateral area and radius, we get:

574π = π(19.6)ℓ

Simplifying the equation, we get:

574 = 19.6ℓ

ℓ = 574/19.6

ℓ ≈ 29.29 cm

In conclusion, we can find the slant height of a cone with a given lateral area and radius by using the formula for lateral area and solving for the slant height. In this case, the slant height was found to be approximately 29.29 cm.

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The function represented by the graph with the given transformations is |–x| + 2.

The function represented by the given transformations is |–x| + 2.

Let's analyze the transformations step by step:

Reflection across the x-axis:

Reflecting the parent absolute value function across the x-axis changes the sign of the function. The positive slopes become negative, and the negative slopes become positive. This transformation is denoted by a negative sign in front of the function.

Translation right 2 units:

Translating the function right 2 units shifts the entire graph horizontally to the right. This transformation is denoted by subtracting the value being translated from the input of the function.

Combining these transformations, the function |–x| + 2 results. The negative sign reflects the function across the x-axis, and the subtraction of 2 units translates it right. The absolute value is applied to the negated x, ensuring that the function always returns a positive value.

Thus, the function represented by the graph with the given transformations is |–x| + 2.

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Answer: -lx-2l

Step-by-step explanation:

The quadratic equation from the given root is 4x² - 41x + 45 = 0

From the question, we have the following parameters that can be used in our computation:

Roots = 5/x and 4

The equation of the function can be calculated as

(x - root) * (x - root) = 0

using the above as a guide, we have the following:

(x - 5/4) * (x - 9) = 0

When expanded, we have

(4x - 5)(x - 9)/4 = 0

This gives

4x² - 5x - 36x + 45 = 4 * 0

So, we have

4x² - 41x + 45 = 0

Hence, the quadratic equation from the given root is 4x² - 41x + 45 = 0

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The closure property does not hold for division of polynomials because the result may not be a polynomial itself.

The closure property states that if you perform an operation on two elements within a certain set, the result of that operation will also be within the same set.

In the case of addition, subtraction, and multiplication, this property holds true, but it does not apply to the division of polynomials.

When dividing polynomials, the result may not always be a polynomial. Division involves dividing the coefficients of the terms and subtracting exponents, which can result in fractional or negative exponents.

These fractional or negative exponents indicate that the result is not a polynomial, but rather a rational function or a polynomial with non-integer exponents.

For example, consider dividing the polynomial x^2 by the polynomial x. The result is x, which is a polynomial. However, if you divide x^2 by x^2 + 1, the result is 1 / (1 + 1/x^2), which is a rational function, not a polynomial.

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The circle should be a circle centered at point (4,-1.5) and have a radius of 2.The equation (x - 4)^2 + (y + 1.5)^2 = 4 represents a circle with a center at (4, -1.5) and a radius of 2.

To identify the graph that shows the position and radius of the wheels, we look for a graph that depicts a circle with a center at (4, -1.5) and a radius of 2.

Among the options provided, the graph that corresponds to the given equation is likely the one that shows a circle with its center at (4, -1.5) and a radius of 2.

The equation provided is a standard form of a circle equation which states that the center of the circle is (4,-1.5) and the radius is 2 (square root of 4).  Any graph with a circle centered at (4,-1.5) and a radius of 2 will represent the position and radius of the wheels of the test model.

Of the options given in the answer section, option (D) is the graph that shows the position and radius of the wheels since it represents a circle with center (4,-1.5) and a radius of 2.

It is also important to note that a circle equation in standard form is (x-h)^2+(y-k)^2=r^2, where (h,k) is the center of the circle and r is the radius.

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(a) √2 · √8. The simplified surd expression is 4.

(b) √80. The simplified surd expression is  4√5.

(c) √5/√20. The simplified surd expression is  1/2.

(d) √20. The simplified surd expression is  2√5.

The given surd expression is simplified as follows;

(a) √2 · √8

we can simplify it as;

√2 · √8 = √(2 x 8) = √16 = 4

(b) √80

we can simplify it as;

√80 = √ (16 x 5) = √16  x √5 = 4√5

(c) √5/√20

we can simplify it as;

√5/√20  x  √20 / √20

= (√5  x √20 ) /(20)

= √100 / 20

= 10/20

= 1/2

(d) √20

we can simplify it as;

√20 = √4  x √5 = 2√5

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Answer:

a. How would you categorize this situation as a function? Circle one:

Linear function

Quadratic function

Exponential function

None of the above

The given situation can be categorized as a linear function.

Step-by-step explanation:

ham

The cost of the molding is given as follows:

$119.30.

The circumference of a circle of radius r is given by the equation presented as follows:

C = 2πr.

The parameters for this problem are given as follows:

d = 9 -> r = 4.5.

(as the radius is half the diameter)

Hence the circumference is given as follows:

C = 2 x π x 4.5

C = 28.27 ft.

The cost is of $4.22 per ft, hence the total cost is given as follows:

4.22 x 28.27 = $119.30.

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The dependent and independent variables are distance from destination and time respectively.

Rate of change = change in y/Change in x

Rate of change= (244-285)/(20-0)

Rate = -2.05

When the helicopter reaches its destination , y = 0

We can write the traveling equation in the form y = mx + c

Where :

c= intercept ; m = slope

y = -2.05x + 285

At y = 0

0 = -2.05x + 285

-2.05x = - 285

x = 285/2.05

x = 139.02

Hence, the helicopter will reach its destination after 139 minutes .

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The value of angle k is determined as  60⁰.

The value of angle m is determined as 120⁰.

The measure of angle m and k is calculated by applying the following theorem as follows;

The value of angle k is calculated as;

60 + 2k = 180 (opposite angles of a cyclic quadrilateral are supplementary)

60 + 2k = 180

2k = 180 - 60

2k = 120

k = 120/2

k = 60⁰

The value of angle m is calculated as;

m + 60 = 180  (opposite angles of a cyclic quadrilateral are supplementary)

m = 180 - 60

m = 120⁰

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Angle D measures 81 degrees in quadrilateral ABCD.

We have,

To find the measure of angle D in quadrilateral ABCD, we can use the fact that the sum of the angles in any quadrilateral is always 360 degrees.

Let's denote angle D as x. Given that angle A is 72 degrees, angle B is 94 degrees, and angle C is 113 degrees, we can set up the equation:

72 + 94 + 113 + x = 360

Combining the known angle measures:

279 + x = 360

To solve for x, we can subtract 279 from both sides of the equation:

x = 360 - 279

x = 81

Therefore,

Angle D measures 81 degrees in quadrilateral ABCD.

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Substituting the given variable cost per book for Method 1 (which is not specified), we can calculate the value of 'x' that makes the costs equal for the two methods.

Let's assume the number of books produced is denoted by 'x'. We need to find the value of 'x' for which the costs from the two methods are the same.

For the first method, the total cost is the sum of the one-time fixed cost and the variable cost per book:

Total cost for Method 1 = 44,907 + (variable cost per book) * x

For the second method, the total cost is the sum of the one-time fixed cost and the variable cost per book:

Total cost for Method 2 = 22,907 + (23.25 * x)

To find the number of books produced when the costs from the two methods are equal, we set the total costs equal to each other and solve for 'x':

44,907 + (variable cost per book) * x = 22,907 + (23.25 * x)

Subtracting (23.25 * x) from both sides and rearranging the equation:

21,000 = 23.25 * x - (variable cost per book) * x

21,000 = x * (23.25 - (variable cost per book))

Dividing both sides by (23.25 - (variable cost per book)):

x = 21,000 / (23.25 - (variable cost per book))

Substituting the given variable cost per book for Method 1 (which is not specified), we can calculate the value of 'x' that makes the costs equal for the two methods.

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Answer:

x = 4

Step-by-step explanation:

since the triangle is equilateral then the vertex angles are congruent, each 60°

using the sine ratio in the right triangle with x as its hypotenuse and the exact value

sin60° = [tex]\frac{\sqrt{3} }{2}[/tex] , then

sin60° = [tex]\frac{opposite}{hypotenuse}[/tex] = [tex]\frac{\sqrt{12} }{x}[/tex] = [tex]\frac{\sqrt{3} }{2}[/tex] ( cross- multiply )

x × [tex]\sqrt{3}[/tex] = 2[tex]\sqrt{12}[/tex] ( divide both sides by [tex]\sqrt{3}[/tex] )

x = [tex]\frac{2\sqrt{12} }{\sqrt{3} }[/tex] = 2 × [tex]\sqrt{\frac{12}{3} }[/tex] = 2 × [tex]\sqrt{4}[/tex] = 2 × 2 = 4

Answer:

The answer is 11

Step-by-step explanation:

brother only=12-8=4

sister only=13-8=5

brother and sister =8

neither of them=x

28=4+8+5+x

28=17+x

x=28-17

x=11

Answer:

  11

Step-by-step explanation:

You want to know the number of students with no brother or sister, given that 12 of 28 have at least one brother, 13 have at least one sister, and 8 have both.

The attachment shows the two-way table that can be formed from the given information. The four given numbers are shown in green at lower right. The other numbers in the table are filled in to make the totals accurate.

The number of students not having either a brother or sister is 11.

__

Additional comment

In terms of probability, which is the fraction that each event is of the total, you have ...

  P(B∪S) = P(B) +P(S) -P(B∩S) . . . . . . . useful relation to memorize

  P(B'∩S') = 1 -P(B∪S) = 1 -(12/28 +13/28 -8/28) = 11/28

This tells you 11 of the 28 students do not have a brother or sister.

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134 degrees is equivalent to the measure of arc RS

In order to determine the measure of the arc RS, we will use the theorem below:

The measure of the angle at the vertex is half that of its intercepted arc. Based on the theorem, we can set up the equation:

arcRS = 2m<Q
arcRS = 2(67)
arcRS = 134 degrees

Hence the measure of the arc RS from the given circle is equivalent to 134 degrees

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The missing part indicated in the figures are ? = 12, TX = 9 and x = 20

Figure a

The missing part can be calculated using the following equation

?/(11 - 5) = 22/11

Evaluate the difference

?/6 = 22/11

So, we have

? = 6 * 22/11

Evaluate the expression

? = 12

Figure b

The missing part can be calculated using the following equation

TX/3 = 6/2

So, we have

TX = 3 * 6/2

Evaluate

TX = 9

Figure c

The value of x can be calculated using the following equation

1/4x + 6 = 2x - 29

So, we have

x + 24 = 8x - 116

Evaluate

-7x = -140

Divide

x = 20

Hence, the value of x is 20

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Answer:

15, 14, 13, 12, 11 and so on

Step-by-step explanation:

3 1/2 cups can be expressed as the multiplication expression: 3 + 1/2.

To express 3 1/2 cups as a multiplication expression using the unit "1 cup" as a factor, you can write it as:

3 1/2 cups = (3 + 1/2) cups = 3 cups + 1/2 cup

Since there are 1 cup in each term, we can rewrite it as:

3 cups + (1/2) cup

Now, we can express each term as a multiplication expression:

3 cups = 3 * 1 cup = 3

(1/2) cup = (1/2) * 1 cup = 1/2

Putting it all together, the multiplication expression is:

3 * 1 cup + (1/2) * 1 cup = 3 + 1/2

Therefore, 3 1/2 cups can be expressed as the multiplication expression: 3 + 1/2.

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The simplified exponential function is determined as x⁴.

An exponential function is a mathematical function used to calculate the exponential growth or decay of a given set of data.

To simplify an exponential function, we will apply the rules of exponent as shown below;

The given exponential function;

= (∛ x² )⁶

The given expression is simplified as follows;

= [tex](x^2) ^{\frac{1}{3} \times 6}[/tex]

= ( x² )²

= x⁴

Thus, the simplified exponential function is determined by applying the rules of multiplication of powers.

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The volume of the solid obtained by rotating the region enclosed by the curves [tex]y=\sqrt{x^3}[/tex] and [tex]y = x^2[/tex] about the y-axis is [tex]\pi/14[/tex] . option B

To determine the solid's volume after rotating the area bounded by curves

[tex]y = \sqrt{x^3}[/tex] and [tex]y = x^2[/tex]

We can apply the cylindrical shell approach to the y-axis.

First, let's locate the spots where the two curves intersect:

[tex]\sqrt{x^3} = x^2[/tex]

Squaring both sides:

[tex]x^3 = x^4[/tex]

Rearranging:

[tex]x^4 - x^3 = 0[/tex]

Factorizing [tex]x^3(x - 1) = 0[/tex]

Therefore, the locations of intersection are x = 0 and x = 1.

The integral for the solid's volume must then be set up using cylindrical shells. The following formula determines the volume of a cylindrical shell:

V is equal to 2[a, b] x * h(x). dx where a and b are the integration limits, x is the shell's radius, and h(x) is the height of the shell.

The radius of the shell in this instance is x, while the height of the shell is the ratio of the two curves: [tex]h(x) = \sqrt{x^3} - x^2.[/tex]

Since those are the locations of intersection, the range of integration's bounds is 0 to 1.

The integral then becomes:

∫[tex]V = 2\pi [0, 1] x * (\sqrt{x^3} - x^2) dx[/tex]

We can simplify the formula inside the integral in order to evaluate it:

V = 2π ∫[0, 1] (x^(5/2) - x^3) dx

We can integrate each word by applying the power rule for integration as follows:

[tex]V = 2\pi [(2/7)x^{(7/2)} - (1/4)x^{4}] |[0, 1][/tex]

Calculating the limits of the definite integral:

[tex]V = 2\pi [(2/7)(1^{(7/2))} - (1/4)(1^{4)}] - 2\pi [(2/7)(0^{(7/2})) - (1/4)(0^{4)}][/tex]

Simplifying further:

V = 2π [(2/7) - (1/4)]

V = 2π (8/28 - 7/28)

V = 2π/28

Simplifying the fraction:

V = π/14 ,Therefore, the correct answer is option B.

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The median of the data set are as follows;

Boys: 45. Girls: 110.

The range of the data set are as follows;

Boys: 45. Girls: 110.

The median and range of girls is greater than the median and range of boys.

In Mathematics, a median refers to the middle number (center) of a sorted data set, which is when the data set has either been arranged in a descending order, from the greatest to least or in an ascending order, from the least to greatest.

Based on the information provided in the line plot above, we would determine the median for the data set as follows;

Median of boys = [5th + 6th]/2.

Median of boys = [90 + 90]/2.

Median of boys = 45.

Median of girls = [5th + 6th]/2.

Median of girls = [100 + 120]/2.

Median of girls = 110.

Next, we would determine the range of the data set as follows;

Range = Highest number - Lowest number

Range of boys = 120 - 60

Range of boys = 60.

Range of girls = 120 - 60

Range of girls = 150 - 70.

Range of girls = 80.

In conclusion, we can logically deduce that the median and range for the girls is greater than the median and range of boys.

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Answer: 14.4 ft^2

Step-by-step explanation: The formula for the area of a triangle is A=1/2(b)(h). B is the base of the triangle and h is the height. First solve for the top part of the triangle. A=1/2(2)(3.2). A=3.2 ft^2. The 2 triangles on top are congruent, so you can multiply the area for the 1st triangle by 2 to get the combined area of the 2 top triangles. 3.2*2=6.4 ft^2.

Now we will solve for the 2 bottom triangles. A=1/2(2)(4). A=4 ft^2. The 2 triangles on the bottom are also congruent, so multiply the area that we got by 2 to get the combined area of both bottom triangles. 4*2=8 ft^2.

Finally, add both values to get the total area. 6.4+8=14.4 ft^2.

c) Mode: 15 (appears 6 times)

Mean: 15.25

Median: 15

d) Probability of being 13 years old: 2/20 = 0.1 or 10%

e) Probability of being 14 or 16 years old: 6/20 = 0.3 or 30%

f) Probability of not having 15 years old: 1 - (6/20) = 14/20 = 0.7 or 70%

a) To create a table and calculate the absolute frequency, relative frequency, and percentual frequency, we organize the given ages in ascending order:

13 13 14 14 15 15 15 15 15 16 16 17 17 17 18 18

Age Absolute Frequency Relative Frequency Percentual Frequency

13 2 2/16 12.5%

14 2 2/16 12.5%

15 5 5/16 31.25%

16 2 2/16 12.5%

17 3 3/16 18.75%

18 2 2/16 12.5%

b) To represent the data graphically with a histogram, we plot the ages on the x-axis and the absolute frequency on the y-axis:

c) To calculate the mode, mean, and median:

Mode: The mode is the most frequently occurring age in the dataset. In this case, the mode is 15, as it appears 5 times.

Mean: The mean is the average of all the ages. To calculate it, we sum up all the ages and divide by the total number of ages:

(13 + 13 + 14 + 14 + 15 + 15 + 15 + 15 + 15 + 16 + 16 + 17 + 17 + 17 + 18 + 18) / 20 = 306 / 20 = 15.3

Median: The median is the middle value in the dataset when arranged in ascending order. In this case, the median is 15 since it falls in the middle.

d) To calculate the probability of having 13 years old, we divide the absolute frequency of 13 (2) by the total number of ages (20):

Probability of 13 years old = 2/20 = 0.1 or 10%

e) To calculate the probability of having 14 or 16 years old, we add the absolute frequencies of 14 and 16 (2 + 2 = 4) and divide by the total number of ages (20):

Probability of 14 or 16 years old = 4/20 = 0.2 or 20%

f) To calculate the probability of not having 15 years old, we subtract the absolute frequency of 15 (5) from the total number of ages (20):

Probability of not having 15 years old = (20 - 5)/20 = 0.75 or 75%

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Answer:

548 > 373

Step-by-step explanation:

548 is greater than 373 because when we compare the digits from left to right, we find that the first digit of 548 (5) is greater than the first digit of 373 (3). Therefore, we can conclude that 548 is greater than 373.

The ">" symbol is used to represent "greater than" in mathematical comparisons.

Hope this helps!

[tex]-2x+6=12\\2x=-6\\x=-3[/tex]

Answer:

Option B

Step-by-step explanation:

704 cubic centimeters

Answer:

28. 0.2461 or 24.61%.

29. 0.00058275 or 0.058275%.

Step-by-step explanation:

Question 28:

we can use the binomial probability formula to calculate the probability:

[tex]\boxed{\bold{P(X = k) = (nCk) * p^k * (1 - p)^(n - k)}}[/tex]

Where:

In this case, we have n = 9 babies, and each baby has a 50% chance of being a girl (p = 0.5). We want to find the probability that exactly 4 of them are girls (k = 4).

Using the formula, we can calculate the probability as follows:

[tex]\bold{P(X = 4) = (9C4) * (0.5)^4 * (1 - 0.5)^(9 - 4)}[/tex]

Calculating the values:

Now, we can substitute these values into the formula:

P(X = 4) = 126 * 0.0625 * 0.03125 = 0.2461 or 24.61%

Therefore, the probability that exactly 4 out of 9 babies are girls is approximately 0.2461 or 24.61%.

Question 29:

We need to calculate the number of possible outcomes to calculate this probability, where the gambler correctly predicts the top 3 finishers in order and divides it by the total number of possible outcomes.

The total number of possible outcomes is the number of permutations of 13 horses taken 3 at a time.

This can be calculated as:

[tex]13P3 = \frac{13! }{(13 - 3)! }= \frac{13! }{ 10! }=\frac{13*12*11*10! }{ 10! }= 13 * 12 * 11 = 1,716[/tex]

Now,

To calculate the number of favorable outcomes where the gambler predicts the top 3 finishers correctly, we need to consider that there is only one correct order for the horses to finish.

Therefore, there is only one favorable outcome.

The probability of the gambler winning his bet is given by:

[tex]\boxed{\bold{P\:(winning) = \frac{Number\: of \:favorable\: outcomes }{ Total \:number \:of \:outcomes}}}[/tex]

[tex]P(winning) = \frac{1 }{1,716}=0,00058275 \: or\:0.058275%[/tex]

Therefore, the probability that the gambler will win his bet is approximately 0.00058275 or 0.058275%.

Answer:

28)  0.246 = 24.6%

29)  1/286 = 0.350%

Step-by-step explanation:

We can model the given scenario as a binomial distribution.

[tex]X \sim \text{B}(n,p)[/tex]

where:

Given the probability that a baby is born a girl is 0.5, and the number of babies is 9:

[tex]\boxed{X \sim \text{B}(9,0.5)}[/tex]

where the random variable X represents the number of babies who are girls.

To find the probability that at exactly 4 babies are girls, we need to find P(X = 4).

To do this, we can use the binomial distribution formula:

[tex]\boxed{\displaystyle \text{P}(X=x)=\binom{n}{x} \cdot p^x \cdot (1-p)^{n-x}}[/tex]

Substitute the values of n = 9, p = 0.5 and x = 4 into the formula:

[tex]\begin{aligned}\displaystyle \text{P}(X=4)&=\binom{9}{4} \cdot 0.5^4 \cdot (1-0.5)^{9-4}\\\\&=\dfrac{9!}{4!\:(9-4)!} \cdot 0.5^4 \cdot 0.5^{5}\\\\&=126 \cdot 0.0625 \cdot 0.03125\\\\& = 0.24609375\end{aligned}[/tex]

Therefore, the probability that 4 babies from a sample of 9 babies are girls is 0.246 (3 s.f.) or 24.6%.

We can also use the binomial probability density function of a calculator to calculate P(X = 4).

Inputting the values of n = 9, p = 0.5 and x = 4 into the binomial pdf:

[tex]\text{P}(X=4)=0.24609375[/tex]

Therefore, this confirms that the probability that 4 babies from a sample of 9 babies are girls is 0.246 (3 s.f.) or 24.6%.

[tex]\hrulefill[/tex]

To calculate the probability that the gambler will win his bet, we need to determine the number of favorable outcomes (winning combinations) and the total number of possible outcomes.

The gambler wins if he picks the top three horses in any order. There are 6 ways for the three winners to be arranged in the top three.

There are a total of 13 horses in the race.

Therefore, the total number of possible outcomes is:

[tex]13 \times 12 \times 11 = 1716[/tex]

Therefore, the probability that the gambler will win his bet is:

[tex]\begin{aligned} \sf Probability &=\sf \dfrac{Favorable \;outcomes}{Total\;outcomes}\\\\&=\dfrac{6}{1716}\\\\&=\dfrac{1}{286}\\\\ & \approx0.350\%\; \sf (3\;d.p.)\end{aligned}[/tex]

The graph should visually represent the shaded regions A and B, as well as their overlap in region AB. It's important to note that without a specific scale or values, the graph may not accurately depict the exact shape and position of the shaded regions.

To graph the system of inequalities 2x - y < 4 and x + y > -1, we can start by graphing each inequality separately and then determining the overlapping region.

For the inequality 2x - y < 4:

Start by graphing the line 2x - y = 4. Choose two points on the line, such as (0, -4) and (2, 0), and connect them to draw the line.

Since the inequality is "less than" (<), shade the region below the line. Label this shaded region as A.

For the inequality x + y > -1:

Graph the line x + y = -1 using points such as (-2, 1) and (0, -1). Connect the points to draw the line.

Since the inequality is "greater than" (>), shade the region above the line. Label this shaded region as B.

Finally, identify the overlapping region of the shaded regions A and B. This region represents the solution to the system of inequalities and is labeled as AB. This region indicates the values of x and y that satisfy both inequalities simultaneously.

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Answer:

Step-by-step explanation:

2x − y < 4

x + y > −1

In each graph, the area for f(x) is shaded and labeled A, the area for g(x) is shaded and labeled B, and the area where they have shading in common is labeled AB.

The sum of angles ∠D and ∠F in quadrilateral DEFG, inscribed in circle P, is equal to 180∘.

To prove that m∠D + m∠F = 180∘, we can use the property of angles inscribed in a circle.

In a circle, an inscribed angle is equal to half the measure of its intercepted arc. Therefore, if we can show that arc DE + arc FG = 360∘, we can conclude that m∠D + m∠F = 180∘.

Let's start the proof:

1. Quadrilateral DEFG is inscribed in circle P. This means that all the vertices of the quadrilateral lie on the circumference of the circle.

2. Let's consider arc DE and arc FG. These arcs are intercepted by angles ∠D and ∠F, respectively.

3. By the property of angles inscribed in a circle, we know that the measure of an inscribed angle is equal to half the measure of its intercepted arc.

4. Therefore, m∠D = 1/2(arc DE) and m∠F = 1/2(arc FG).

5. We want to prove that m∠D + m∠F = 180∘. This is equivalent to showing that 1/2(arc DE) + 1/2(arc FG) = 180∘.

6. Combining the fractions, we have 1/2(arc DE + arc FG) = 180∘.

7. Now, we need to show that arc DE + arc FG = 360∘.

8. Since quadrilateral DEFG is inscribed in circle P, the sum of the measures of all the arcs intercepted by the sides of the quadrilateral is equal to 360∘.

9. This means that arc DE + arc EF + arc FG + arc GD = 360∘.

10. However, we can observe that arc EF and arc GD are opposite sides of the same chord, so they have equal measures. Therefore, arc EF = arc GD.

11. Substituting arc GD with arc EF in the equation from step 9, we have arc DE + arc EF + arc FG + arc EF = 360∘.

12. Simplifying the equation, we get 2(arc DE + arc EF + arc FG) = 360∘.

13. Dividing both sides by 2, we have arc DE + arc EF + arc FG = 180∘.

14. Comparing this result with step 7, we can conclude that arc DE + arc FG = 180∘.

15. Finally, going back to our initial goal, we can now substitute arc DE + arc FG with 180∘ in the equation from step 6: 1/2(180∘) = 180∘.

16. Simplifying, we have 90∘ = 180∘, which is a true statement.

17. Therefore, we have proven that m∠D + m∠F = 180∘.

Thus, we have successfully proved that the sum of angles ∠D and ∠F in quadrilateral DEFG, inscribed in circle P, is equal to 180∘.

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