Q: A horse owner wants to construct a rectangular fence next to a barn that is 40 feet long, using the barn as one side of the fence. If the owner has 80 feet of fencing available, what is the maximum area that can be enclosed?

$$A = 400$$\text{ ft}^2$$ when the width is 20$ ft

Q: A rectangular corral is to be built using a barn wall as one side and $$80$$ feet of fencing for the other three sides. What are the dimensions that maximize the area?

$$20 ft$$ $$ \times $$ $$40 ft$$

Question 1

A horse owner wants to construct a rectangular fence next to a barn that is 40 feet long, using the barn as one side of the fence. If the owner has 80 feet of fencing available, what is the maximum area that can be enclosed?

Accepted Answer

$$A = 400$$	ext{ ft}^2$$ when the width is 20$ ft

Question 2

Given the scenario of the horse owner and the fence, which equation correctly expresses the area $$A(x) of $$the corral as a function of $$x$$, the width perpendicular to the barn?

Accepted Answer

$$A(x) = x(80 - 2x)$$

Question 3

A gardener has a $$20 ft by$$ $$10 ft $$rectangular plot and wants to build a walkway of uniform width $$x$$ around the garden, leaving $$240$$ square feet for planting. Which equation models the situation?

Accepted Answer

$$(20 - 2x)(10 - 2x) = 240$$

Question 4

Solve the equation $$x^2 + 5x - 36 = 0$. What are the solutions?

Accepted Answer

$$x = 4,\ x = -9$$

Question 5

Which of the following is the correct polynomial with zeros at $$-1$$, $$4$$, and $$7$$?

Accepted Answer

$$P(x) = (x + 1)(x - 4)(x - 7)$$

Question 6

If $$P(x) = x^3$ + 2x^2$ - x - 2$$, what is $$P(3)$$?

Accepted Answer

$$P(3) = 40$$

Question 7

A rectangular garden is $$20 ft by$$ $$10 ft. If a $$walkway of width $$x is $$built around it, and the remaining area is $$240 ft$^2$$, what is the value of $$x$$?

Accepted Answer

$$x = 2$$

Question 8

Given the equation $$2x^2 + 5x - 7 = 0$, which of the following is the correct solution using the quadratic formula?

Accepted Answer

$$x = \frac{-5 \pm \sqrt{25 + 56}}{4}$$

Question 9

Which of the following best describes the process of synthetic division?

Accepted Answer

A shortcut method for dividing a polynomial by a linear binomial of the form $$x - c$$

Question 10

If $$P(x) = x^3$ + 2x^2$ - x - 2$$, use synthetic division to find $$P(1). $$What is the result?

Accepted Answer

$$0$$

Question 11

A rectangular corral is to be built using a barn wall as one side and $$80$$ feet of fencing for the other three sides. What are the dimensions that maximize the area?

Accepted Answer

$$20 ft$$ $$ 	imes $$ $$40 ft$$

Question 12

Which of the following is the correct factored form of $$x^2 + 5x - 36$?

Accepted Answer

$$(x + 9)(x - 4)$$

Question 13

Given $$P(x) = 2x^3$ - 3x^2$ - 19x + 18$$, what is the value of $$P(2)$$?

Accepted Answer

$$-10$$

Question 14

If a polynomial $$P(x)$$ has zeros at $$x = -2$$, $$x = 3$$, and $$x = 5$$, which of the following is $$P(x)$$?

Accepted Answer

$$P(x) = (x + 2)(x - 3)(x - 5)$$

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