91Ó°ÊÓ

Find the constant of proportionality, \(k,\) for the given conditions. a. \(y=k x^{3},\) and \(y=64\) when \(x=2\). b. \(y=k x^{3 / 2},\) and \(y=96\) when \(x=16\). c. \(A=k r^{2},\) and \(A=4 \pi\) when \(r=2\). d. \(v=k t^{2},\) and \(v=-256\) when \(t=4\).

Assume \(y\) is inversely proportional to the cube of \(x\). a. If \(x\) doubles, what happens to \(y\) ? b. If \(x\) triples, what happens to \(y\) ? c. If \(x\) is halved, what happens to \(y ?\) d. If \(x\) is reduced to one-third of its value, what happens to \(y\) ?

The radius of Earth is about 6400 kilometers. Assume that Earth is spherical. Express your answers to the questions below in scientific notation. a. Find the surface area of Earth in square meters. b. Find the volume of Earth in cubic meters. c. Find the ratio of the surface area to the volume.

If the radius of a sphere is \(x\) meters, what happens to the surface area and to the volume of a sphere when you: a. Quadruple the radius? b. Multiply the radius by \(n\) ? c. Divide the radius by 3 ? d. Divide the radius by \(n\) ?

Assume a box has a square base and the length of a side of the base is equal to twice the height of the box. a. If the height is 4 inches, what are the dimensions of the base? b. Write functions for the surface area and the volume that are dependent on the height, \(h\). c. If the volume has increased by a factor of \(27,\) what has happened to the height? d. As the height increases, what will happen to the ratio of (surface area)/volume?

A box has volume \(V=\) length \(\cdot\) width \(\cdot\) height. a. Find the volume of a cereal box with dimensions of length \(=19.5 \mathrm{~cm},\) width \(=5 \mathrm{~cm},\) and height \(=27 \mathrm{~cm} .\) (Be sure to specify the unit.) b. If the length and width are doubled, by what factor is the volume increased? c. What are two ways you could increase the volume by a factor of 4 and keep the height the same?

The volume, \(V,\) of a cylindrical can is \(V=\pi r^{2} h\) and the total surface area, \(S\), of the can is $$ \begin{aligned} S &=\text { area of curved surface }+2 \cdot \text { (area of base) } \\ &=2 \pi r h+2 \pi r^{2} \end{aligned} $$ where \(r\) is the radius of the base and \(h\) is the height. a. Assume the height is three times the radius. Write the volume and the surface area as functions of the radius. b. As \(r\) increases, which grows faster, the volume or the surface area? Explain.

The volume, \(V,\) of a cylinder, with radius \(r\) and height \(h,\) is given by the formula \(V=\pi r^{2} h .\) Describe what happens to \(V\) under the following conditions. a. The radius is doubled; the radius is tripled. b. The height is doubled; the height is tripled. c. The radius \(r\) is multiplied by \(n,\) where \(n\) is a positive integer. d. The height is multiplied by \(n,\) where \(n\) is a positive integer.