The height of a right circular cone is decreased by 6 percent. Find the percentage that the radius of the base must be decreased by so that the volume decreases by 20 percent. Recall that the formula for the volume of a right circular cone is
V =
πr2h 3
V=1/3 PI r^2 h
V'=1/2 PI r'^2 h'
v'=.8v, h'=.94h
v'/v=.8= (r'/r)^2 *.94
r'/r= sqrt(.8/.94)=.92
so r must be reduced 8 percent
To find the percentage by which the radius of the base must be decreased, we need to first understand the relationship between the height, radius, and volume of a right circular cone.
The formula for the volume of a right circular cone is given as:
V = (1/3) * π * r^2 * h,
where V represents the volume, r represents the radius of the base, and h represents the height of the cone.
Now, let's solve the problem step by step:
1. Given that the height of the cone is decreased by 6 percent, we can calculate the new height as follows:
New height = original height - decrease percent * original height
= h - (6/100) * h
= h - 0.06h
= 0.94h
2. We know that the volume of the cone decreases by 20 percent. So, we can calculate the new volume as follows:
New volume = original volume - decrease percent * original volume
= V - (20/100) * V
= V - 0.2V
= 0.8V
3. Substitute the given formulas and values into the original volume equation:
0.8V = (1/3) * π * (r^2) * (0.94h)
= (0.94/3) * π * (r^2) * h
4. Cancel out the common factor (0.94/3) and simplify the equation:
0.8V = (0.94/3) * π * (r^2) * h
Divide both sides by (0.94/3) * π * h:
(r^2) = (0.8V) / [(0.94/3) * π * h]
r^2 = (0.8 * 3 * V) / (0.94 * π * h)
r^2 = (2.4V) / (0.94πh)
5. Take the square root of both sides to solve for r:
r = √[(2.4V) / (0.94πh)]
6. Now, we can substitute the original volume equation to express r in terms of the original radius (r) and height (h):
r = √[(2.4 * (1/3) * π * r^2 * h) / (0.94πh)]
r = √[(2.4/3) * r^2]
r = √(0.8 * r^2)
r = √0.8 * r
r = 0.894427191 * r
7. Finally, calculate the percentage by which the radius must be decreased:
Decrease percentage = (1 - new radius / original radius) * 100
= (1 - 0.894427191) * 100
= 0.105572809 * 100
≈ 10.56%
Therefore, the radius of the base must be decreased by approximately 10.56% to decrease the volume by 20%.