Question

the following diagram shows a toy spinning top made up of a plastic cylinder and a plastic cone (tip).
the diameter of the cylinder is 0.6 cm.
the height of the cylinder is 2.5 cm.
the radius of the base of the cone is 2 cm.
the slant height of the cone is 2.8 cm.

(a) (i) calculate the vertical height of the cone, in cm. give your answer correct to three significant figures.
(ii) use your answer from the part (i) to calculate the total volume of the spinning top. give your answer correct to two decimal places.

one spinning top was deemed as defected after the manufacturing process due to insufficient amount of plastic. as a result, the stem was shorter than usual.

(b) find the height of this short stem, given that the amount of plastic used for this spinning top was $8.5\\ \text{cm}^3$.

Explanation:

Step1: Use Pythagoras for cone height

Let vertical height of cone = $h$. Slant height $l=2.8$ cm, base radius $r=2$ cm.
$$h = \sqrt{l^2 - r^2} = \sqrt{2.8^2 - 2^2}$$
$$h = \sqrt{7.84 - 4} = \sqrt{3.84} \approx 1.95959 \text{ cm}$$
Round to 3 significant figures: $h \approx 1.96$ cm

Step2: Calculate cylinder volume

Cylinder diameter = 0.6 cm, so radius $r_c = \frac{0.6}{2} = 0.3$ cm, height $h_c=2.5$ cm.
Volume of cylinder $V_c = \pi r_c^2 h_c = \pi \times 0.3^2 \times 2.5$
$$V_c = \pi \times 0.09 \times 2.5 = 0.225\pi \approx 0.70686 \text{ cm}^3$$

Step3: Calculate cone volume

Use $h=1.96$ cm, $r=2$ cm.
Volume of cone $V_{cone} = \frac{1}{3}\pi r^2 h = \frac{1}{3}\pi \times 2^2 \times 1.96$
$$V_{cone} = \frac{1}{3}\pi \times 4 \times 1.96 = \frac{7.84}{3}\pi \approx 8.22265 \text{ cm}^3$$

Step4: Total volume of spinning top

$$V_{total} = V_c + V_{cone} \approx 0.70686 + 8.22265 = 8.92951 \text{ cm}^3$$
Round to 2 decimal places: $V_{total} \approx 8.93$ cm³

Step5: Find short stem height

Let short stem height = $h_s$. Total plastic used = 8.5 cm³.
$$V_{cone} + \pi r_c^2 h_s = 8.5$$
Substitute $V_{cone} \approx 8.22265$, $r_c=0.3$:
$$8.22265 + \pi \times 0.3^2 \times h_s = 8.5$$
$$0.09\pi h_s = 8.5 - 8.22265 = 0.27735$$
$$h_s = \frac{0.27735}{0.09\pi} \approx \frac{0.27735}{0.28274} \approx 0.98 \text{ cm}$$

Answer:

(a)(i) $\boldsymbol{1.96}$ cm
(a)(ii) $\boldsymbol{8.93}$ cm³
(b) $\boldsymbol{0.98}$ cm