Question

英语数学题求解？

求详细步骤和说明

Answer 1

(a)每秒钟注入的咖啡体积为150/60=2.5mL
然后这点咖啡倒入的是
直径8cm的咖啡杯
底面积=4^2*pi=16pi cm^2
所以每一秒升高距离=体积/底面积=2.5/(16pi) cm/s=5/(32pi) cm/s

(b)过滤器中每秒要流出2.5mL
那么与a)同理速度=体积/底面积=2.5/(6^2*pi)=2.5/(72pi)=5/(144pi) cm/s
那个8cm是糊弄人的

Answer 2

(a) Because the draining speed is a constant, and the mug is cylindrical, the rate of the coffer level in the mug is also a constant, and equals to the draining rate devided by the area of the mug's base:
r = 150/[π(d/2)²] = 150/[π(8/2)²] ≈ 2.98 cm/min

(b) Because the filter is conical, the rate of changes in the coffee level is not a constant.
Let r(t) be the radius of surface of coffee, h the depth of coffee at t second:
r/R = h/H, where R is the radius of the cone (6 cm), and H the height of the cone (12 cm)

r(t) = Rh/H
The volume of coffee at this moment is:
V(t) = (1/3)πr²h(t) = (1/3)π[Rh(t)/H]²h(t) = πR²h³(t)/(3H²)
V'(t) = 3πR²h²(t)/(3H²) = πR²h²(t)h'(t)/H²
h'(t) = H²V'(t)/[πR²h²(t)]
When h(t) = 8 cm, V'(t) = 150 mL/min
h'(t) = 12²*150/(3.14*6²*8²) = 2.99 cm/min