MATHEMATICS HIGH SCHOOL

Pam has 90 m of fencing to enclose an area in a petting zoo with two dividers to separate three types of young animals. The three pens are to have the same area. Express the area function for the three pens in terms of x. Determine the domain and range for the area function

Answers

Answer 1

Answer:

Answer:

The area function is

$A=(135)/(2)x-(9)/(2)x^2$ .

The domain and range of A is $(0,15m)$ and $(0, 253.125 m^2]$ .

Step-by-step explanation:

The given length of fencing is $90 m$ .

Let the length and width of each pen be $x$ and $y$ respectively as shown in the figure.

As there are 3 pens, so, the total area,

$A= 3 xy \;\cdots (i)$

From the figure the total length of fencing is $6x+4y$ .

Here, for a significant area for the animals, $x>0$ as well as $y>0$ as $x$ and $y$ are the sides of ben.

From the given value:

$6x+4y=90\;\cdots (ii)$

$\Rightarrow y=\frac {45}{2}-(3x)/(2)$

Now, from equation (i)

$A=3x\left(\frac {45}{2}-(3x)/(2)\right)$

$\Rightarrow A=(135)/(2)x-(9)/(2)x^2\;\cdots (iii)$

This is the required area function in the terms of variable $x$ .

For the domain of area function, from equation (ii)

$x=15-(2y)/(3)$

$\Rightarrow x<15 m$ [as y>0]

So, the domain of area function is $(0,15m)$ .

For the range of area function:

As $x \rightarrow 0$ or $y\rightarrow 0$ , then $A\rightarrow 0$ [from equation (i)]

$\Rightarrow A>0$

Now, differentiate the area function with respect to $x$ .

$\frac {dA}{dx}=(135)/(2)-9x$

Equate $\frac {dA}{dx}$ to zero to get the extremum point.

$\frac {dA}{dx}=0$

$\Rightarrow (135)/(2)-9x=0$

$\Rightarrow x=(15)/(2)$

Check this point by double differentiation

$\frac {d^2A}{dx^2}=-9$

As, $\frac {d^2A}{dx^2}<0$ , so, point $x=(15)/(2)$ is corresponding to maxima.

Put this value back to equation (iii) to get the maximum value of area function. We have

$A=(135)/(2)* \frac {15}{2}-(9)/(2)* \left(\frac {15}{2}\right)^2$

$\Rightarrow A=253.125 m^2$

Hence, the range of area function is $(0, 253.125 m^2]$ .

Answer 2

Answer:

Final answer:

The area of each pen can be expressed as A(x) = x * (90 - 2x) / 3. The domain of this function is 0 < x < 45, and the range is 0 < A(x) < 300

Explanation:

In this problem, since Pam has to divide the petting zoo into three parts, we can consider the width of each pet pen to be x and the total length of the three pens to be (90 - 2x)/3, given that the total fence is 90m and we have two fences that are x meters long separating the pens. So, the area, A of each pen can be expressed as a function of x: A(x) = x * (90 - 2x) / 3. The domain of this function, or the possible values of x, would be all the values that make the area positive, which are 0 < x < 45. For the range of the function, we analyze the quadratic function which will have a maximum value at x = 15, as the area will be largest when the space is divided evenly, so the maximum area is A(15)= 15 * (60) / 3 = 300. Therefore, the range of the function is 0 < A(x) < 300.

Learn more about Area and Functions here:

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Find the product of (3.7 × 104) and 2

Answers

"769.6" I hope it helps

im assuming the "and 2" part means to also multiply so just multiply 3.7*104 to get 384.8. then multiply the product by 2 and the answer will be 769.6.

Which set of numbers can represent the lengths of the sides of a right triangle? Round to the nearest whole number. 4, 4, 4 4, 6.93, 8 11.2, 16.2, 19.2

Answers

Answer with explanation:

A set of numbers that is triplet is said to form the sides of a right triangle, if it satisfies Pythagorean Theorem,which states that

Square of longest side is equal to sum of Squares of other two sides.

1.First triplet=4,4,4

As, 4²≠4²+4²

Hence it does not form Pythagorean triplet.

2. Second triplet=4, 6.93, 8

Largest side length =8

8²=64

Sum of squares of two smaller sides =(6.93)²+4²

=48.0249+16

=64.0249

=64

3. Third triplet=11.2, 16.2, 19.2

Largest side length =19.2

(19.2)²=368.64

Sum of squares of two smaller sides =(11.2)²+(16.2)²

=125.44+262.44

=387.88

=388(approx)

Option B ⇒4,6.93, 8

Hello,

A: 4 , 4 , 4 not right triangle

B: 4 , 6.93 , 8 or 4²+6.93²=64,0249 ≈64=8²

C: 11.2 , 16.2 , 19.2 or 11.2²+16.2²=387,88 =(19,69...)²

The best choice is B

What is the value of the expression "eight more than three times the difference of four and a number" when n =3?
□44
□19
□17
□11

Answers

Answer:

Step-by-step explanation:

3 (4 - n) + 8

When n = 3:

3 (4 - 3) + 8

3 (1) + 8

3 + 8

Answer:

Step-by-step explanation:

In the SSS statement, _____.A) all three sides of one triangle are equal to the three sides of another triangle

B) two triangles are similar if any sides of one triangle are congruent to any two sides of another triangles

C) two triangles are similar if corresponding sides form a proportion

D) two triangles are congruent if corresponding sides form a proportion

Answers

i believe its c, sorry if im wrong.

Which is closest to 0.5?
0.4, 0.505, 0.449, 0.48 or 0.53?

Answers

Let us calculate the nearness of all the numbers provided
Number        Nearnes
0.400                100
0.505                   5
0.449                  51
0.530                30

Thus 0.505 is the closest number to 0.5 in decimal terms

$1)|0.5-0.4|=0.1\n 2)|0.5-0.505|=0.005\n 3)|0.5-0.48|=0.02\n 4)|0.5-0.53|=0.03$

As you see the seconds difference is the smallest therefore 0.505 is closest to .0.5

Solve the equation and chech -8x+11=11