The base of a solid is a circular disk with radius 3. Find the volume of the solid if parallel cross-sections is perpendicular to the base are isosceles right triangles with hypotenuse lying along the base.
nvm i got it answer is 36!!
To find the volume of the solid, we need to integrate the areas of the cross-sections along the height of the solid.
Let's denote the height of the solid as h. At any given height h, the cross-section is an isosceles right triangle with one of its legs lying along the base.
The length of the leg along the base is equal to the radius of the circular disk, which is 3.
The hypotenuse of the triangle is also given to lie along the base. So, the length of the hypotenuse is also 3.
Now, we can find the length of the other leg of the triangle using the Pythagorean theorem. Since it is an isosceles right triangle, the other leg will also be equal to the length of the hypotenuse.
Using the Pythagorean theorem, we have:
leg^2 + leg^2 = hypotenuse^2
leg^2 + leg^2 = 3^2
2 * leg^2 = 9
leg^2 = 9 / 2
leg = √(9 / 2)
Therefore, at height h, the area of the cross-section is (1/2) * (3) * (√(9 / 2)) = (3/2) * (√(9 / 2))
To find the volume, we need to integrate this area from 0 to the height h.
Volume = ∫(0 to h) (3/2) * (√(9 / 2)) dh
Integrating, we have:
Volume = (3/2) * (√(9 / 2)) * h (evaluated from 0 to h)
Volume = (3/2) * (√(9 / 2)) * h - (3/2) * (√(9 / 2)) * 0
Volume = (3/2) * (√(9 / 2)) * h
Therefore, the volume of the solid is (3/2) * (√(9 / 2)) * h.
To find the volume of the solid, we need to determine the formula for the volume of a shape with isosceles right triangle cross-sections. Let's break down the problem step by step:
1. Imagine a cross-section of the solid.
2. Since the cross-section is an isosceles right triangle, we can draw two equal legs perpendicular to the base.
3. The hypotenuse of each triangle lies along the circular base, meaning it is the radius of the disk.
To find the volume of the solid, we need to integrate the area of each cross-section over the entire height of the solid.
4. Let's define the height of the solid as 'h' (from the base to the top).
Now, let's find the formula for the area of each cross-section:
The base of each cross-section is a right-angled triangle. The length of one leg is equal to the radius of the circular disk, which is 3 units. Let's call this leg length 'x'.
Given that the cross-section is an isosceles right triangle, the other leg will also have length 'x'.
The hypotenuse of each triangle is also equal to the radius of the circular base, which is 3 units.
5. Using the Pythagorean theorem, we can find the length of the legs:
x^2 + x^2 = 3^2
2x^2 = 9
x^2 = 9/2
x = √(9/2)
6. Now we have the area of each cross-section:
Area = 0.5 * base * height
= 0.5 * x * x
= 0.5 * (√(9/2)) * (√(9/2))
= 0.5 * 9/2
= 9/4
7. To find the volume of the solid, we need to integrate this area over the entire height of the solid, which is 'h':
Volume = ∫(0 to h) (9/4) dh
= (9/4) * ∫(0 to h) dh
= (9/4) * [h] (from 0 to h)
= (9/4) * (h - 0)
= (9/4) * h
8. Finally, we substitute the height of the solid, which is 'h', with its actual value:
Volume = (9/4) * h
Volume = (9/4) * h, where h is the height of the solid.
Therefore, the volume of the solid with a circular base of radius 3 and isosceles right triangle cross-sections is given by the formula (9/4) * h, where 'h' is the height of the solid.