6.5.3 Volume of Three Dimensional Shapes, PT3 Practice

Question 8:
A spherical balloon expands such that its volume increases from 36π cm³ to 288π cm³ .
Find the increment of the radius, in cm, of the balloon.

Solution:
$\begin{array}{l}{V}_1=\frac{4}{3}\pi r_1{}^3\\ \frac{4}{3}\pi r_1{}^3=36\pi \\ r_1{}^3=36\pi \times \frac{3}{4\pi }\\ r_1{}^3=27\\ r_1=3\text{ cm}\\ \\ V_2=\frac{4}{3}\pi r_2{}^3\\ \frac{4}{3}\pi r_2{}^3=288\pi \\ r_2{}^3=288\pi \times \frac{3}{4\pi }\\ r_2{}^3=216\\ r_2=6\text{ cm}\\ \\ \text{Increment}=r_2-r_1\\ =6-3\\ =3\text{ cm}\end{array}$

Question 9:
A boy is given three pieces of cards by his teacher. The cards consist of one rectangle card and two circle cards of the same size as shown in the diagram below. The circumference of each circle is 66 cm.

He is required to combine the cards to form a right circular cylinder.
Calculate the volume, in cm³, of the right circular cylinder.

Solution:
$\begin{array}{l}2\pi r=66\\ 2\times \frac{22}{7}\times r=66\\ r=\frac{66\times 7}{2\times 22}\\ =10.5\text{ cm}\\ \\ \text{Volume}=\pi r^{2}h\\ =\frac{22}{7}\times {10.5}^2\times 20\\ =6930{\text{ cm}}^3\end{array}$

Question 10:
Diagram below shows cube P and cube Q with the total surface area of 726 cm² and 1014 cm² respectively.

Find the difference in volume between cube P and cube Q.

Solution:
$\begin{array}{l}\text{Area of one surface area of cube }P=\frac{726}{6}=121{\text{ cm}}^2\\ x\left(\text{Length}\right)\times x\left(\text{Width}\right)=121\\ x^2=121\\ x=11\\ \\ \text{Area of one surface area of cube }Q=\frac{1014}{6}=169{\text{ cm}}^2\\ y\left(\text{Length}\right)\times y\left(\text{Width}\right)=169\\ y^2=169\\ y=13\\ \\ \text{Difference in volume}\\ ={13}^3-{11}^3\\ =2197-1331\\ =866{\text{ cm}}^3\end{array}$