Solve the given problem.

A piece of cardboard is twice as long as it is wide. It is to be made into a box with an open top by cutting two-inch squares from each corner and folding up the sides. Let $x$ represent the width of the original piece of cardboard.

(a) Represent the length of the original piece of cardboard in terms of $x$.

(b) What will be the dimensions of the bottom rectangular base of the box? Give the restrictions on $x$.

(c) Determine a function $V$ that represents the volume of the box in terms of $x$.

(d) For what dimensions of the bottom of the box will the volume be $320$ cubic inches? Determine analytically and support graphically.

(e) Determine graphically (to the nearest tenth of an inch) the values of $x$ if the box is to have a volume between $400$ and $500$ cubic inches.

One measurement used in judging kite-flying competitions is the size of the angle formed by the kite string and the ground. This angle can be used to find the height of the kite. Suppose the length of the string is 600 feet and the angle at which the kite is flying measures $40^{\circ}$. Calculate the height, h, of the kite.

Factor using the formula for the sum or difference of two cubes.

$$64x^3 + 27$$

What prevents a plant cell from bursting in a hypotonic solution?

Solve this equation analytically for all complex solutions, giving exact forms in your solution set. Then graph the left side of the equation as $\mathrm{Y}_1$ in the suggested viewing window and, using the capabilities of your calculator, support the real solutions.

$x^4-15 x^2-16=0$; $[-5,5]$ by $[-100,100]$

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

$$\sqrt{-100}$$

Use an end behavior diagram (the ones shown in the discussion) to describe the end behavior of the graph of each function. Do not use a calculator.

$$P(x)=12 x^{107,4 y y}$$

Which of the following describes a property of pure water?

• changes temperature rapidly
• acidic pH
• molecules are noncohesive
• can hold many molecules in solution
• more dense as a solid

Find a polynomial function $P(x)$ having leading coefficient one, least possible degree, real coefficients, and the given zeros.

$6$ and $-2$

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 this equation analytically for all complex solutions, giving exact forms in your solution set. Then graph the left side of the equation as $\mathrm{Y}_1$ in the suggested viewing window and, using the capabilities of your calculator, support the real solutions.

$4 x^4-29 x^2+25=0$; $[-5,5]$ by $[-50,100]$

What is ATP and what is its role in the cell?

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

$$x^2=-32$$

Answer the following given problems.

Suppose that a polynomial function $P$ is defined in such a way that $P(3)=-4$ and $P(4)=-10$. Can we be certain that there is no zero between $3$ and $4$? Explain, using a graph.