Chapter 7

In Exercises \(1-4\), write a formula for \(y\) in terms of \(x\) if \(y\) satisfies the given conditions. Proportional to the \(5^{\text {th }}\) power of \(x,\) and \(y=744\) when \(x=2\)

In Exercises \(1-21,\) solve the equation for the variable. $$ x^{3}=50 $$

In Exercises \(1-5,\) identify the exponent and the coefficient for each power function. The area of a square of side \(x\) is \(A=x^{2}\).

In Exercises \(1-4\), write a formula for \(y\) in terms of \(x\) if \(y\) satisfies the given conditions. Proportional to the cube of \(x\), with constant of proportionality -0.35 .

In Exercises \(1-21,\) solve the equation for the variable. $$ 2 x^{2}=8.6 $$

Are the functions power functions? $$ y=12 \cdot 14^{x} $$

Identify the exponent and the coefficient for each power function. The perimeter of a square of side \(x\) is \(P=4 x\).

In Exercises \(1-4\), write a formula for \(y\) in terms of \(x\) if \(y\) satisfies the given conditions. Proportional to the square of \(x,\) and \(y=1000\) when \(x=5\)

In Exercises \(1-21,\) solve the equation for the variable. $$ 4=x^{-1 / 2} $$

Are the functions power functions? $$ y=3 x^{3}+2 x^{2} $$

Identify the exponent and the coefficient for each power function. The side of a cube of volume \(V\) is \(x=\sqrt[3]{V}\).

In Exercises \(1-4\), write a formula for \(y\) in terms of \(x\) if \(y\) satisfies the given conditions. Proportional to the \(4^{\text {th }}\) power of \(x,\) and \(y=10.125\) when \(x=3\).

In Exercises \(1-21,\) solve the equation for the variable. $$ 4 w^{3}+7=0 $$

Are the functions power functions? $$ y=2 /\left(x^{3}\right) $$

Find a formula for \(s\) in terms of \(t\) if \(s\) is proportional to the square root of \(t,\) and \(s=100\) when \(t=50\).

In Exercises \(1-21,\) solve the equation for the variable. $$ z^{2}+5=0 $$

Are the functions power functions? $$ y=x^{3} / 2 $$

Identify the exponent and the coefficient for each power function. The surface area of a sphere of radius \(r\) is \(S=4 \pi r^{2}\).

If \(A\) is inversely proportional to the cube of \(B,\) and \(A=20.5\) when \(B=-4,\) write \(A\) as a power function of \(B\)

In Exercises \(1-21,\) solve the equation for the variable. $$ 2 b^{4}-11=81 $$

Are the functions power functions? $$ y=\sqrt{4 x^{4}} $$

The area, \(A,\) of a rectangle whose length is 3 times its width is given by \(A=3 w^{2}\), where \(w\) is its width. (a) Identify the coefficient and exponent of this power function. (b) If the width is \(5 \mathrm{~cm}\), what is the area of the rectangle?

Suppose \(c\) is directly proportional to the square of \(d\). If \(c=50\) when \(d=5,\) find the constant of proportionality and write the formula for \(c\) in terms of \(d\). Use your formula to find \(c\) when \(d=7\).

In Exercises \(1-21,\) solve the equation for the variable. $$ \sqrt{a}-2=7 $$

In Exercises \(7-16\), write the expression as a constant times a power of a variable. Identify the coefficient and the exponent. $$ 3 \sqrt{p} $$

The volume, \(V\), of a cylinder whose radius is 5 times its height is given by \(V=\frac{1}{5} \pi r^{3}\), where \(r\) is the radius. (a) Identify the coefficient and exponent of this power function. (b) If the radius is \(2 \mathrm{~cm},\) what is the volume? (c) If the height is \(0.8 \mathrm{~cm}\), what is the volume?

Suppose \(c\) is inversely proportional to the square of \(d\). If \(c=50\) when \(d=5\), find the constant of proportion-ality and write the formula for \(c\) in terms of \(d\). Use your formula to find \(c\) when \(d=7\).

In Exercises \(1-21,\) solve the equation for the variable. $$ 3 \sqrt[3]{x}+1=16 $$

Write the expression as a constant times a power of a variable. Identify the coefficient and the exponent. $$ \sqrt{2 b} $$

In Exercises \(9-12,\) write a formula representing the function. The strength, \(S\), of a beam is proportional to the square of its thickness, \(h .\)

In Exercises \(1-21,\) solve the equation for the variable. $$ \sqrt{y-2}=11 $$

Write the expression as a constant times a power of a variable. Identify the coefficient and the exponent. $$ \frac{4}{\sqrt[4]{z}} $$

In Exercises \(9-12,\) write a formula representing the function. The energy, \(E\), expended by a swimming dolphin is proportional to the cube of the speed, \(v\), of the dolphin.

In Exercises \(1-21,\) solve the equation for the variable. $$ \sqrt{2 y-1}=9 $$

Write the expression as a constant times a power of a variable. Identify the coefficient and the exponent. $$ \frac{\sqrt{x}}{\sqrt[3]{x}} $$

In Exercises \(9-12,\) write a formula representing the function. The radius, \(r,\) of a circle is proportional to the square root of the area, \(A\).

In Exercises \(1-21,\) solve the equation for the variable. $$ \sqrt{3 x-2}+1=10 $$

Write the expression as a constant times a power of a variable. Identify the coefficient and the exponent. $$ \frac{x^{4}}{4 \sqrt{x^{2}}}, x>0 $$

The radius, \(r,\) of a circle is proportional to the square root of the area, \(A\). Kinetic energy, \(K\), is proportional to the square of velocity, \(v .\)

In Exercises \(1-21,\) solve the equation for the variable. $$ (x+1)^{2}+4=29 $$

Write the expression as a constant times a power of a variable. Identify the coefficient and the exponent. $$ \left(\frac{1}{5 \sqrt{x}}\right)^{3} $$

In Exercises \(13-16,\) what happens to \(y\) when \(x\) is doubled? Here \(k\) is a positive constant. $$ y=k x^{3} $$

In Exercises \(1-21,\) solve the equation for the variable. $$ (3 c-2)^{3}-50=100 $$

Write the expression as a constant times a power of a variable. Identify the coefficient and the exponent. $$ \sqrt[3]{\frac{8}{x^{6}}} $$

In Exercises \(13-16,\) what happens to \(y\) when \(x\) is doubled? Here \(k\) is a positive constant. $$ y=\frac{k}{x^{3}} $$

In Exercises \(1-21,\) solve the equation for the variable. $$ 2 x=54 x^{-2} $$

Write the expression as a constant times a power of a variable. Identify the coefficient and the exponent. $$ (2 \sqrt{x}) x^{2} $$

In Exercises \(14-17\) (a) Is \(y\) proportional, or is it inversely proportional, to a positive power of \(x\) ? (b) Make a table of values showing corresponding values for \(y\) when \(x\) is \(1,10,100,\) and 1000 . (c) Use your table to determine whether \(y\) increases or decreases as \(x\) gets larger. \(y=2 x^{2}\)

In Exercises \(13-16,\) what happens to \(y\) when \(x\) is doubled? Here \(k\) is a positive constant. $$ x y=k $$

In Exercises \(1-21,\) solve the equation for the variable. $$ 2 p^{5}+64=0 $$