An ecology center wants to set up an experimental garden using three hundred meters of fencing to enclose a rectangular area of five thousand square meters.

(a) Let $x$ meters represent the width of the garden. Why must the length be $150-x$ meters?

(b) Write an expression $\mathscr{A}(x)$ that represents the area of the garden.

(c) What are the restrictions on $x$ ?

(d) Use the given values for area and length of fencing to find the dimensions of the garden.

In this exercise, use a calculator to find the coordinates of the vertex of the graph. Express: coordinates to the nearest hundredth. Consider the function

$$P(x)=-2.64 x^2+5.47 x+3.54 .$$

Solve the equation or inequality analytically, and support your result graphically.

$$x^2-x-3>0$$

Why did Lincoln issue the Emancipation Proclamation.

Write the given number in simplest form, without a negative radicand.

$$\sqrt{-100}$$

Solve the given equation. For equations with real solutions, support your answers graphically.

$$(3 x-1)^2=12$$

Solve the given equation. For equations with real solutions, support your answers graphically.

$$x^2=-32$$

Determine whether the given statement is true or false. If it is false, tell why.

A complex number might not be a pure imaginary number.

Solve the given problem.

A cylindrical aluminum can is being constructed to have a height $h$ of four inches. If the can is to have a volume of twenty-eight cubic inches, approximate its radius $r$. (Hint: $V=\pi r^2 h$.)

In this exercise, graph the function in the standard window of a calculator, and use the root-finding capabilities to solve the equation $P(x)=0$. Express solutions as approximations to the nearest hundredth. Consider the function

$$P(x)=-2.64 x^2+5.47 x+3.54 .$$

For the given complex number, (a) state the real part, (b) state the imaginary part, and (c) identify the number as one or more of the following: real, pure imaginary, or nonreal complex.

$$3 i$$

In this exercise, use the discriminant to explain how to determine the number of $x$-intercepts the graph of $P$ will have before graphing it on your calculator. Consider the function

$$P(x)=-2.64 x^2+5.47 x+3.54 .$$

Match the given equation in Column I with its solution(s) in Column II. Do not use a calculator.

$$\begin{array}{lll} &\text{I}&\text{II}\\ & x^2=-4 &\text{A. }\pm 2 i\\ & &\text{B. } \pm 2 \sqrt{2}\\ & &\text{C. }\pm i \sqrt{2}\\ & &\text{D. }2\\ & &\text{E. }\pm \sqrt{2}\\ & &\text{F. }-2\\ & &\text{G. }\pm 2\\ & &\text{H. }\pm 2 i \sqrt{2}\\ \end{array}$$

Perform the operation. Write answers in standard form.

$$i(5+i)(5-i)$$