AMC 8 2019 Problem 9
AMC8 2019 年第 9 题
立体几何★☆☆☆☆
Alex and Felicia each have cats as pets. Alex buys cat food in cylindrical cans that are 6 cm in diameter and 12 cm high. Felicia buys cat food in cylindrical cans that are 12 cm in diameter and 6 cm high. What is the ratio of the volume of one of Alex's cans to the volume of one of Felicia's cans?
Alex 和 Felicia 他们都有宠物猫。Alex 买的宠物猫粮是装在底面直径为 6 厘米，高度 12 厘米的圆柱形罐子里的。Felicia 买的猫粮则装在底面直径是 12 厘米，高为 6 厘米的圆柱形罐子里。问 Alex 的罐子的体积和 Felicia 的罐子的体积之比是多少?
- A.1:4
- B.1:2
- C.1:1
- D.2:1
- E.4:1
正确答案 · Correct Answer
B
解析 · Solution
Solution 1
Using the formula for the volume of a cylinder , we get Alex, , and Felicia, . We can quickly notice that cancels out on both sides and that Alex's volume is 1/2 of Felicia's leaving 1/2 = 1:2 as the answer. edits by Octopus72 LaTeX corrections by themathkidwoohooooo
Using the formula for the volume of a cylinder, we get that the volume of Alex's can is , and that the volume of Felicia's can is . Now, we divide the volume of Alex's can by the volume of Felicia's can, so we get , which is . -(Algebruh123)2020
The ratio of the numbers is 1/2. Looking closely at the formula , we see that the will cancel, meaning that the ratio of them will be = . -Lcz
The second can is size in each of 2 dimensions, and size in 1 dimension. So the volume ratio is , i.e. .
Without calculating much, you can do ( <-- which is Alex's volume, with ra being Alex's radius ( diameter), and $ha$ being her cylinders height <-- which is Felicia's volume, with $rf$ being Felicia's radius, and $hf$ being her cylinders height. Since we need the ratio between Alexa's and Felicias, we can do The cancel out, then substitute back in the numbers, which gives you: -wahahaqueenie
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