Q # ..18 I need your help

Mathematics

1 answer:

andriy [413]3 years ago

8 0

Answer: y = -5x - 3.

Explanation:

When you know the slope and the y-intercept you can directly write the slope-intercept equation of the line.

The slope-intercept equation of the line has the form: y = mx + b, where m is the slope and b is the y-intercept.

Here you know m = -5 and b = -3, hence you just have to substitute those values in the general slope-intercept form, which is:

y = -5x - 3.

That equation represents the line, which means tha every point in the line will be an ordered pair of the function. For example:

x y = -5x - 3

- 10 -5(-10) - 3 = 50 - 3 = 47

0 0 - 3 = - 3 (this is the y-intercept)

10 -5(10) - 3 = - 50 - 3 = - 53

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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

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