#3 A solid has a base in the form of the ellipse: x^2/25 + y^2/16 = 1. Find the volume if every cross section perpendicular to the x-axis is an isosceles triangle whose altitude is 6 inches.

#4 Use the same base and cross sections as #3, but change the axis to the y-axis.

Let's just do the quarter of it over the right half 0 < x < 5 and multiply by 2 at the end

that means the triangle has base 2y and height 6 everywhere so its Area is (1/2)(2y)(6) = 6y and a slice of volume is 6ydx
we need to integrate that from x = 0 to x = 5 (and not forget to double the answer)
y^2 = 16(1 - x^2/25)
y = 4 (1 - x^2/25)^.5 (just use + root)
so we have
24 integral(1 - x^2/25)^.5 dx
or
(24/5) integral (25 - x^2)^.5 dx
=12/5 [ x sqrt(25-x^2)+25sin^-1(x/5) ] }
(looked up sqrt(p^2-x^2) dx)
from x = 0 to x = 5
at x = 5
12/5 [ 0 + 25 sin^-1(1) ]= 12/5[25 pi/2]
=30 pi
at x = 0
12/5[ 0 + 0] = 0
so
30 pi now times 2 = 60 pi
for heavens sake check my arithmetic

It is an ellipse on the bottom, pi has to be involved in the volume I think.

Ah, yes. I forgot the √ in my integral.

I also get 60π

I thought it strange that there was no π, but I was in a hurry at the time.

Score one more for Damon.

A simpler solution uses the fact that the area of an ellipse is πab

Here, a=4 and b=5, so the area of the ellipse is 20π.

Since all the triangles have the same height (6), the volume is (1/2)(6)(20π) = 60π

#3 Alright, let's calculate the volume of this solid that has an elliptical base and isosceles triangle cross sections perpendicular to the x-axis. We need to find the length and width of each cross section first.

Now, since the altitude of each triangle is given as 6 inches, we can consider the base of each triangle as the x-axis. Since the cross sections are isosceles triangles, the y-coordinate of the apex can be obtained from the equation of the ellipse when x is equal to the length of the base of the triangle.

Using the equation of the ellipse, x^2/25 + y^2/16 = 1, when x equals the base length, we can solve for y to find the height of the triangle. Let me grab my trusty calculator...

Calculating... calculating... ah, here it is! The formula for the height of the triangle is h = 4√(25 - (25/16)x^2).

Now, let's find the area of each cross section. Since the triangles are isosceles, we can use the formula for the area of a triangle: A = (1/2)bh. In this case, the base length is x, and we found that the height is h = 4√(25 - (25/16)x^2).

Now all we need to do is integrate the area of each cross section along the x-axis to find the volume. But oh dear, it seems I've out-joked myself with all this math talk. Perhaps I should have told a triangle pun instead?

To find the volume of a solid with a given base and cross sections, we need to integrate the area of each cross section.

#3: Cross sections perpendicular to the x-axis
In this case, each cross section perpendicular to the x-axis is an isosceles triangle with an altitude of 6 inches. To determine the base of the triangle, we need to find the width of the ellipse at each level.

The equation of the ellipse is given by:

x^2/25 + y^2/16 = 1

To find the width at a specific y-value, we can solve the equation for x. Rearranging the equation, we get:

x^2 = 25 - (25/16) * y^2
x = sqrt(25 - (25/16) * y^2)

Now we can determine the area of each cross section by multiplying the base (width) by the altitude (6 inches).
The area of each cross section is given by:

A = (base * altitude)/2

Therefore, the volume of the solid can be calculated by integrating the area over the range of y-values that the ellipse covers.

V = integral [from -4 to 4] [(6 * sqrt(25 - (25/16) * y^2))/2] dy

Solving this integral will give us the volume of the solid.

#4: Cross sections perpendicular to the y-axis
The process for finding the volume is similar, but instead of integrating with respect to y, we will integrate with respect to x since the cross sections are perpendicular to the y-axis.

Using the same ellipse equation:

x^2/25 + y^2/16 = 1

To find the width of each cross section at a specific x-value, we can solve the equation for y:

y^2 = 16 - (16/25) * x^2
y = sqrt(16 - (16/25) * x^2)

The area of each cross section is still given by:

A = (base * altitude)/2

Thus, the volume of the solid can be calculated by integrating the area over the range of x-values that the ellipse covers.

V = integral [from -5 to 5] [(6 * sqrt(16 - (16/25) * x^2))/2] dx

By solving this integral, we can find the volume of the solid when the cross sections are perpendicular to the y-axis.

using symmetry, we can see that since the area of each triangle is 1/2 yz

v = 2∫[0,5] yz dx
where y = 4√(1-x^2/25) and z=6
v = 48/5∫[0,5] ∫(25-x^2) dx = 800