I don't know how to do this I am lost

Nataly [62]

Answer:4/10

Step-by-step explanation:

6 0

2 years ago

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 t

yanalaym [24]

Answer:

A

Step-by-step 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:

$\displaystyle A=\frac{\sqrt{3}}{4}s^2$

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:

$\displaystyle V=\int_0^1\frac{\sqrt3}{4}(y)^2\, dx$

Substitute:

$\displaystyle V=\int_0^1\frac{\sqrt3}{4}(e^x)^2\, dx$

Evaluate the integral. Simplify:

$\displaystyle V=\frac{\sqrt3}{4}\int_0^1e^{2x}\, dx$

Integrate using u-substitution:

$\displaystyle V=\frac{\sqrt3}{8}\left(e^{2x}\Big|_0^1\right)$

Evaluate:

$\displaystyle V=\frac{\sqrt3}{8}\left(e^{2(1)}-e^{2(0)} \right)$

Therefore, the volume of the solid is:

$\displaystyle V=\frac{\sqrt3}{8}\left(e^2-1\right)$

Our answer is A.

3 0

2 years ago

Read 2 more answers

F(t) = 200(0.95)^t What is the rate of change?

Nataly [62]

Answer:

f(t)=-5%

Step-by-step explanation:

4 0

2 years ago

Read 2 more answers

Student council expects $150 students for a dance and spends $10 on food for each student. The council expects the number of stu

Natalija [7]

Answer:

$A) S(t)=150(1.08)^{t-1}$ $B) A(t)=10(0.96)^{t-1}$

$C) If\:t = 1\\E(t)=150\times10\\ \\If\:t >1\\E(t)=150(1.08)^{t-1}\times10(0.96)^{t-1}$

Step-by-step explanation:

Considering that The Student Council expects to 150 students for a dance, and spends $10 per student.

If the expectancy is that the number of students increases by 8%

Then since the growth is 8%

we can write the the Predicted Number of Students as a Geometric Sequence,

A.

$a_{1}=150\\a_{2}=150\times 1.08 =162\\a_{3}=162\times 1.08= 174.96\approx 175\\a_{4}=175\times1.08=189\\a_{5}=189\times 1.08=204.12\approx 204\\$

Therefore,

$S(t)=150(1.08)^{t-1}$

B) If we spend $10 per student on the 1st year, but the Council wants to reduce the expenditure 4% yearly. Since we want to reduce 4% we must multiply by 96%.

$b_{1}=10\\b_{2}=10*(0.96)=9.6\\b_{3}=9.6*(0.96)=9.22\\b_{4}=9.22*(0.96)=8.85\\b_{5}=8.85*(0.96)=8.50$