Related rates with spheres, circles, and cubes

2 The volume of a spherical balloon is increasing at a constant rate of \(50 \mathrm {~cm} ^ { 3 }\) per second. Find the rate of increase of the radius when the radius is 10 cm . [Volume of a sphere \(= \frac { 4 } { 3 } \pi r ^ { 3 }\).]

8. The volume \(V\) of a spherical balloon is increasing at a constant rate of \(250 \mathrm {~cm} ^ { 3 } \mathrm {~s} ^ { - 1 }\). Find the rate of increase of the radius of the balloon, in \(\mathrm { cm } \mathrm { s } ^ { - 1 }\), at the instant when the volume of the balloon is \(12000 \mathrm {~cm} ^ { 3 }\).
Give your answer to 2 significant figures.
[0pt] [You may assume that the volume \(V\) of a sphere of radius \(r\) is given by the formula \(V = \frac { 4 } { 3 } \pi r ^ { 3 }\).]

6. The area \(A\) of a circle is increasing at a constant rate of \(1.5 \mathrm {~cm} ^ { 2 } \mathrm {~s} ^ { - 1 }\). Find, to 3 significant figures, the rate at which the radius \(r\) of the circle is increasing when the area of the circle is \(2 \mathrm {~cm} ^ { 2 }\).
(5)

6. Oil is leaking from a storage container onto a flat section of concrete at a rate of \(0.48 \mathrm {~cm} ^ { 3 } \mathrm {~s} ^ { - 1 }\). The leaking oil spreads to form a pool with an increasing circular cross-section. The pool has a constant uniform thickness of 3 mm . Find the rate at which the radius \(r\) of the pool of oil is increasing at the instant when \(r = 5 \mathrm {~cm}\). Give your answer, in \(\mathrm { cm } \mathrm { s } ^ { - 1 }\), to 3 significant figures. \includegraphics[max width=\textwidth, alt={}, center]{245bbe52-3a14-4494-af17-7711caf79b22-19_104_95_2617_1786}

2. \begin{figure}[h] \end{figure} Figure 1 shows a metal cube which is expanding uniformly as it is heated. At time \(t\) seconds, the length of each edge of the cube is \(x \mathrm {~cm}\), and the volume of the cube is \(V \mathrm {~cm} ^ { 3 }\).

1. Show that \(\frac { \mathrm { d } V } { \mathrm {~d} x } = 3 x ^ { 2 }\) Given that the volume, \(V \mathrm {~cm} ^ { 3 }\), increases at a constant rate of \(0.048 \mathrm {~cm} ^ { 3 } \mathrm {~s} ^ { - 1 }\),
2. find \(\frac { \mathrm { d } x } { \mathrm {~d} t }\), when \(x = 8\)
3. find the rate of increase of the total surface area of the cube, in \(\mathrm { cm } ^ { 2 } \mathrm {~s} ^ { - 1 }\), when \(x = 8\)

2. \begin{figure}[h] \end{figure} Figure 1 shows a cube which is increasing in size.
At time \(t\) seconds,

• the length of each edge of the cube is \(x \mathrm {~cm}\)
• the surface area of the cube is \(S \mathrm {~cm} ^ { 2 }\)
• the volume of the cube is \(V \mathrm {~cm} ^ { 3 }\)

Given that the surface area of the cube is increasing at a constant rate of \(4 \mathrm {~cm} ^ { 2 } \mathrm {~s} ^ { - 1 }\)

1. show that \(\frac { \mathrm { d } x } { \mathrm {~d} t } = \frac { k } { x }\) where \(k\) is a constant to be found,
2. show that \(\frac { \mathrm { d } V } { \mathrm {~d} t } = V ^ { p }\) where \(p\) is a constant to be found.

1. A balloon is filled with air at a constant rate of \(80 \mathrm {~cm} ^ { 3 }\) per second.

Assuming that the balloon is spherical as it is filled, find to 3 significant figures the rate at which its radius is increasing at the instant when its radius is 6 cm .

2 A spherical balloon of radius \(r \mathrm {~cm}\) has volume \(V \mathrm {~cm} ^ { 3 }\), where \(V = \frac { 4 } { 3 } \pi r ^ { 3 }\). The balloon is inflated at a constant rate of \(10 \mathrm {~cm} ^ { 3 } \mathrm {~s} ^ { - 1 }\). Find the rate of increase of \(r\) when \(r = 8\).

3 A giant spherical balloon is being inflated in a theme park. The radius of the balloon is increasing at a rate of 12 cm per hour. Find the rate at which the surface area of the balloon is increasing at the instant when the radius is 150 cm . Give your answer in \(\mathrm { cm } ^ { 2 }\) per hour correct to 2 significant figures.
[0pt] [Surface area of sphere \(= 4 \pi r ^ { 2 }\).]