# Cups A and B are cone shaped and have heights of 34 cm and 23 cm and openings with radii of 15 cm and 17 cm, respectively. If cup B is full and its contents are poured into cup A, will cup A overflow? If not how high will cup A be filled?

Jan 14, 2018

$32.44$ cm high in cup "A" will be filled.

#### Explanation:

Cup B (Conical): radius and height are r_b=17 cm ; h_b =23 cm

Capcaity of Cup B is ${V}_{b} = \frac{1}{3} \cdot \pi \cdot {r}_{b}^{2} \cdot {h}_{b}$

${V}_{b} = \frac{1}{3} \cdot \pi \cdot {17}^{2} \cdot 23 = 6960.72$ cubic cm

Cup A (Conical): radius and height are r_a=15 cm ; h_a =34 cm

Capcaity of Cup A is ${V}_{a} = \frac{1}{3} \cdot \pi \cdot {r}_{a}^{2} \cdot {h}_{a}$

${V}_{a} = \frac{1}{3} \cdot \pi \cdot {15}^{2} \cdot 34 = 8011.06$ cubic cm

${r}_{a} / {h}_{a} = \frac{15}{34}$ . Since the capacity of cup B is less than that of

cup A , the content will not overflow. Let the height and radius in

the content cone in cup A be ${h}_{2} \mathmr{and} {r}_{2}$

r_2/h_2=15/34 ; r_2= 15/34*h_2 ; V_c=6960.72 ; V_c is the

volume of contents in cup A $\therefore \frac{1}{3} \cdot \pi \cdot {r}_{2}^{2} \cdot {h}_{2} = 6960.72$

${r}_{2}^{2} \cdot {h}_{2} = \frac{6960.72 \cdot 3}{\pi}$ or

${\left(\frac{15}{34} \cdot {h}_{2}\right)}^{2} \cdot {h}_{2} = \frac{6960.72 \cdot 3}{\pi}$ or

${h}^{3} = \frac{6960.72 \cdot 3}{\pi} \cdot {34}^{2} / {15}^{2} = 34150.8$

$\therefore h = \sqrt[3]{34150.8} \approx 32.44 \left(2 \mathrm{dp}\right)$ cm .

$32.44$ cm high in cup "A" will be filled. [Ans]