a pie dish has a base diameter 8 inches and a top diameter 10 inches and a height 2 inches. What is the volume of the dish?
How would I do this problem? I could place it on the origin so that the cross sections would be circles, and the area would then be pir^2 where the base radius would be 4 inches and the top 5 inches..?
I'm assuming the pie dish basically a cone with the top cut off flipped upside down? Filled?
Well, yes, I believe you have it. Your answer should be the same as rotating the shape enclosed by .5x+4, the y-axis, and x=2 about the x axis.
Yes, that's correct! To find the volume of the pie dish, you can think of it as a composition of multiple circular cross-sections stacked on top of each other.
Here's how you can approach the problem step by step:
1. First, visualize the shape of the pie dish. It has a base diameter of 8 inches, which means the base radius is half of that, or 4 inches. The top diameter is 10 inches, so the top radius is half of that, or 5 inches. The height of the dish is given as 2 inches.
2. As you mentioned, placing the dish on the origin makes it easier to visualize and calculate the volumes of the individual circular cross-sections.
3. Now, consider a thin slice of the dish. This thin slice can be thought of as a circular disk. To find the volume of this disk, you can use the formula for the volume of a cylinder: V = πr^2h, where r is the radius and h is the height.
4. Since the cross-sections of the dish are circular, you need to find the volume of each slice separately and then integrate them.
5. Start with the bottom slice, which has a radius of 4 inches and a height of 2 inches.
V1 = π(4^2) * 2
6. Then, move on to the top slice, which has a radius of 5 inches and a height of 2 inches.
V2 = π(5^2) * 2
7. Finally, integrate the volumes of all the slices to find the total volume of the dish:
V_total = ∫[V1 to V2] dV
V_total = ∫[π(4^2) * 2 to π(5^2) * 2] dV
V_total = ∫[32π to 50π] dV ]
V_total = [32πV to 50πV]
= (50π - 32π) cubic inches
= 18π cubic inches
Therefore, the volume of the pie dish is 18π cubic inches.