An ice cream cone has a diameter of 7 cm and its height is 9 cm. It is filled with a scoop of spherical shaped ice cream of radius 3.5 cm.

Find: (Give all answers correct to the nearest whole number)

(a) on melting, is the ice cream sufficient to fill the cone completely without any wastage?

(b) the volume of ice cream, if any, is in excess or less.

Given,

Height of cone(h) = 9 cm

Diameter of cone = 7 cm

Radius of cone (R) = $\dfrac{7}{2}$ = 3.5 cm

Radius of spherical scoop (r) = 3.5 cm

By formula,

$V_{cone} = \dfrac{1}{3}\pi r^2h$

Substituting values we get :

$V_{cone} = \dfrac{1}{3} \times \dfrac{22}{7} \times (3.5)^2 \times (9) \\[1em] = \dfrac{22}{7} \times 12.25 \times 3 \\[1em] = \dfrac{22}{7} \times 36.75 \\[1em] = \dfrac{808.5}{7} \\[1em] = 115.5 \approx 116 \text{ cm}^3.$

By formula,

Volume of sphere = $\dfrac{4}{3} \pi r^3$

$V_{\text{spherical scoop}}= \dfrac{4}{3} \times \dfrac{22}{7} \times (3.5)^3 \\[1em] = \dfrac{4}{3} \times \dfrac{22}{7} \times 42.875 \\[1em] = \dfrac{3773}{21} \\[1em] = 179.67 \approx 180 \text{ cm}^3.$

Since, 180 > 116,

Hence, the ice cream is in excess.

(b) Excess ice cream = Volume of spherical scoop - Volume of cone

= 180 – 116 = 64 cm³.

Hence, excess volume of ice cream = 64 cm³.