Question

The base of a solid is the region enclosed by the graphs of y=e^x, y=0, x=0, and x = 1. If each cross-section perpendicular to the x-axis is an equilateral triangle, then the volume is

(A) sqrt(3)/8(e^2-1)
(B) sqrt(3)/4(e^2-1)
(C) sqrt(3)/2(e^2-1)
PLS EXPLAIN

Answer 1

A.) sqrt(3)/8(e^2-1) is the answer.

Explanation:

#CarryOnLearning

Answer 2

A

Explanation:

The base of a solid in the region enclosed by the graphs of y = eˣ, y = 0, x = 0, and x = 1. Each cross-section perpendicular to the x-axis is an equilateral triangle. We want to find the volume of the solid.

Please refer to the graph below. We are concerned with the red region.

In order to find the volume, we essentially sum up the area of the figure at each x value. So, we integrate from x = 0 to x = 1.

The area for an equilateral triangle is given by:

`\displaystyle A=(√(3))/(4)s^2`

Where s is the side length of the triangle.

Since the triangle lies perpendicular on the region, the side length of the triangle at x is simply y, which is eˣ.

Therefore, our volume is:

`\displaystyle V=\int_0^1(\sqrt3)/(4)(y)^2\, dx`

Substitute:

`\displaystyle V=\int_0^1(\sqrt3)/(4)(e^x)^2\, dx`

Evaluate the integral. Simplify:

`\displaystyle V=(\sqrt3)/(4)\int_0^1e^(2x)\, dx`

Integrate using u-substitution:

`\displaystyle V=(\sqrt3)/(8)\left(e^(2x)\Big|_0^1\right)`

Evaluate:

`\displaystyle V=(\sqrt3)/(8)\left(e^(2(1))-e^(2(0)) \right)`

Therefore, the volume of the solid is:

`\displaystyle V=(\sqrt3)/(8)\left(e^2-1\right)`

Our answer is A.