From a 12-cm by a 12-cm piece of cardboard, square corners are cut out so that the sides can be folded up to make a box.

a) Express the volume of the box as a function of the side x, in centimeters, of a cut-out square.

b) Find the domain of the function.

c) Graph the function with a graphing calculator.

d) What dimensions yield the maximum volume?

An open box is to be made from a rectangular piece of material, 16 inches by 12 inches, by cutting equal squares from the corners and turning up the sides. (a) Write the volume V of the box as a function of x. Determine the domain of the function. (b) Sketch the graph of the function and approximate the dimensions of the box that yield a maximum volume. (c) Find values of such that V=120. Which of these values is a physical impossibility in the construction of the box? Explain. (d) What value of x should you use to make the tallest possible box with a volume of 120 cubic inches?

A rectangle is to be inscribed in a semicircle of radius $8$, with one side lying on the diameter of the circle. What isthe maximum possible area of the rectangle?

Use the Laws of Continuity and Theorems 2 and 3 to show that the function is continuous.

$$f ( x ) = 3 x ^ { 3 } + 8 x ^ { 2 } - 20 x$$

An open box with a square base is to be made from a square piece of cardboard that measures 12 cm on each side. A square will be cut out from each corner of the cardboard and the sides will be turned up to form the box. Find the dimensions that yield the maximum volume.

A car dealership has $24$ $\mathrm{ft}$ of dividers with which to enclose a rectangular play space in a corner of a customer lounge. The sides against the wall require no partition. Suppose the play space is $x$ feet long.

a) Express the area of the play space as a function of $x$.

b) Find the domain of the function.

c) Using the graph shown below, determine the dimensions that yield the maximum area.

Find the domain of each function. $g(x)=\frac{1}{\ln x}$

The Goodyear Blimp can be seen flying at an altitude of 3500 feet above the Motor Speedway during the Indianapolis 500 race. The slanted distance directly to the Pagoda at the start-finish line is d feet. Express the horizontal distance h as a function of d.

A summer camp has $240$ $\mathrm{ft}$ of float line with which to rope off three adjacent rectangular areas of a lake for swimming lessons, one for each of three levels of swimming ability. A beach forms one side of the swimming areas. Suppose that the width of each area is $x$ yards.

a) Express the total area of the three swimming areas as a function of $x$.

b) Find the domain of the function.

c) Using the graph of the function shown below, determine the dimensions that yield the maximum area.

A propane tank is constructed by welding hemispheres to the ends of a right circular cylinder. Write the volume V of the tank as a function of r and l, where r is the radius of the cylinder and hemispheres and l is the length of the cylinder.

An installer is replacing the carpet in a 12-foot by 15-foot living room. The new carpet costs $13.99 per square yard The formula f (x)=9x converts square yards to square feet. How much will the new carpet cost?

Indicate whether the following statements is sometimes, always, or never true. Equations of the form

$$\sqrt { x + a } = b$$

have at least one real solution when a is negative and b is positive

The Goodyear Blimp can be seen flying at an altitude of $3500$ $\mathrm{ft}$ above the Motor Speedway during the Indianapolis $500$ race. The slanted distance directly to the Pagoda at the startfinish line is $d$ feet. Express the horizontal distance $h$ as a function of $d$.

Renée’s Decorating Service recommends putting a border around the top of the four walls in a dining room that is 3 feet longer than it is wide. Find the dimensions of the room if the total length of the border is 54 feet.