Answer to Question #246258 in Calculus for Jess

Question

In exercises 6 – 7, find a formula for the described function and state its domain.
6. The area of a rectangle is 100 m2. Express the perimeter as a function of the length of one
of its sides.
7. A right circular cone is inside a cube. The base of the cone is inscribed in one face of the
cube and its vertex is in the opposite face. Express the volume of the region between the
cone and the cube as a function of the length of the edge of the cube.

Answer 1

Solution.

6.

$S=ab,$ where $a \text{ and }b$ are sides of rectangle, S is square of rectangle.

From here $b=\frac{S}{a}.$ Such as $S= 100$ m^2, then $b=\frac{100}{a}.$

Perimeter of rectangle is

$$P=2(a+b)=2(a+\frac{S}{a})=\frac{2a^2+200}{a}.$$

Answer. $P=\frac{2a^2+200}{a}.$

7.

Let be $a$ is the length of the edge of the cube, $V_1$ is the volume if the cube and $V_2$ Is the volume of the cone.

$$V_1=a^3.$$

The radius of the base of the cone is $r=\frac{a}{2}.$ Then

$$V_2=\frac{1}{3}\pi (\frac{a}{2})^2 a=\frac{1}{12}\pi a^3.$$

So,

$$V=V_1-V_2=a^3-\frac{1}{12}\pi a^3.$$

Answer. $V=a^3-\frac{1}{12}\pi a^3.$