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3 years ago

What should be done to solve the following equation? (7.5)x = 45

Mathematics

2 answers:

Ymorist [56]3 years ago

8 0

Divide each side by 7.5 .

45 divided by 7.5 is 6 .

X equals 6

Tema [17]3 years ago

4 0

Divide each side by 7.5

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3 years ago

Let V be the volume of the solid obtained by rotating about the y-axis the region bounded y = sqrt(25x) and y = x^2/25. Find V b

Westkost [7]

Explanation:

Let $f(x) = \sqrt{25x}$ and $g(x) = \frac{x^2}{25}$ . The differential volume dV of the cylindrical shells is given by

$dV = 2\pi x[f(x) - g(x)]dx$

Integrating this expression, we get

$\displaystyle V = 2\pi\int{x[f(x) - g(x)]}dx$

To determine the limits of integration, we equate the two functions to find their solutions and thus the limits:

$\sqrt{25x} = \dfrac{x^2}{25}$

We can clearly see that x = 0 is one of the solutions. For the other solution/limit, let's solve for x by first taking the square of the equation above:

$25x = \dfrac{x^4}{(25)^2} \Rightarrow \dfrac{x^3}{(25)^3} = 1$

$x^3 =(25)^3 \Rightarrow x = \pm25$

Since we are rotating the functions around the y-axis, we are going to use the x = 25 solution as one of the limits. So the expression for the volume of revolution around the y-axis is

$\displaystyle V = 2\pi\int_0^{25}{x\left(\sqrt{25x} - \frac{x^2}{25}\right)}dx$

$\displaystyle\:\:\:\:=10\pi\int_0^{25}{x^{3/2}}dx - \frac{2\pi}{25}\int_0^{25}{x^3}dx$

$\:\:\:\:=\left(4\pi x^{5/2} - \dfrac{\pi}{50}x^4\right)_0^{25}$

$\:\:\:\:=4\pi(3125) - \pi(7812.5) = 14726.2$

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2 years ago