0000:00:00:00:00Start5Number 5ExplanationA cylindrical packaged drink has a radius that is half of its height. If the full capacity of the package is 729π4 cm3\frac{729\pi}{4} \text{ cm}^34729π cm3. Then the ratio of the volume to the cylindrical lateral surface area is...2:32 : 32:34:94 : 94:99:49 : 49:42:52 : 52:55:25 : 25:2ExplanationFormula Analysis The lateral surface of a cylinder is a rectangle with length 2πr2\pi r2πr and height ttt. Thus, the formula for the lateral surface area is LS=2πrtL_S = 2\pi r tLS=2πrt. We will compare the volume (V=πr2tV = \pi r^2 tV=πr2t) to the lateral surface area (LSL_SLS). VLS=πr2t2πrt\frac{V}{L_S} = \frac{\pi r^2 t}{2\pi r t}LSV=2πrtπr2tVLS=r2\frac{V}{L_S} = \frac{r}{2}LSV=2r Calculating the Ratio In the previous question (Number 4), it was determined that the package height t=9 cmt = 9 \text{ cm}t=9 cm. Since r=12tr = \frac{1}{2}tr=21t, then r=92r = \frac{9}{2}r=29. Substitute the value of rrr into the ratio equation: VLS=922\frac{V}{L_S} = \frac{\frac{9}{2}}{2}LSV=229VLS=94\frac{V}{L_S} = \frac{9}{4}LSV=49 Thus, the ratio is 9:49:49:4.
5Number 5ExplanationA cylindrical packaged drink has a radius that is half of its height. If the full capacity of the package is 729π4 cm3\frac{729\pi}{4} \text{ cm}^34729π cm3. Then the ratio of the volume to the cylindrical lateral surface area is...2:32 : 32:34:94 : 94:99:49 : 49:42:52 : 52:55:25 : 25:2ExplanationFormula Analysis The lateral surface of a cylinder is a rectangle with length 2πr2\pi r2πr and height ttt. Thus, the formula for the lateral surface area is LS=2πrtL_S = 2\pi r tLS=2πrt. We will compare the volume (V=πr2tV = \pi r^2 tV=πr2t) to the lateral surface area (LSL_SLS). VLS=πr2t2πrt\frac{V}{L_S} = \frac{\pi r^2 t}{2\pi r t}LSV=2πrtπr2tVLS=r2\frac{V}{L_S} = \frac{r}{2}LSV=2r Calculating the Ratio In the previous question (Number 4), it was determined that the package height t=9 cmt = 9 \text{ cm}t=9 cm. Since r=12tr = \frac{1}{2}tr=21t, then r=92r = \frac{9}{2}r=29. Substitute the value of rrr into the ratio equation: VLS=922\frac{V}{L_S} = \frac{\frac{9}{2}}{2}LSV=229VLS=94\frac{V}{L_S} = \frac{9}{4}LSV=49 Thus, the ratio is 9:49:49:4.
