Find the volume V of the solid obtained by rotating the region bounded by the given curves about the specified line.
[tex]y = {x}^{2} [/tex]
[tex]y = 4x[/tex]
; about the y-axis

Answers

Answer 1

well, first off, when it comes to volumes by rotation, we'd want to graph them, Check the picture below. Our rotation over the axis will give us a "washer", so we'll be using the washer method.

now, our axis of rotation is the y-axis or namely x = 0, x = 0 is a vertical line, meaning we have to put the functions in y-terms, that is in f(y) form

[tex]y = x^2\implies \pm\sqrt{y}=x\implies \boxed{\pm\sqrt{y}=f(y)} \\\\\\ y = 4x\implies \cfrac{y}{4}=x\implies \boxed{\cfrac{y}{4}=g(y)}[/tex]

if we look at the picture, the parabola is the farthest from the axis of rotation and the line is the closest, or namely R² and r² respectively.

the way I get the area for R² and r², is by using the same I do with "area under the curve", so if I say call the axis of rotation h(y), the way I get the area is

R => f(y) - h(y)

r => g(y) - h(y)

so let's proceed.

[tex]\textit{area under the curve}\\ \begin{array}{llll} \sqrt{y}- 0\implies &\sqrt{y}\\ \frac{y}{4}-0\implies &\frac{y}{4} \end{array}\qquad \qquad \begin{array}{llll} \stackrel{R^2}{(\sqrt{y})^2}-\stackrel{r^2}{\left( \frac{y}{4} \right)^2}\\[-0.5em] \hrulefill\\ y\qquad -\qquad \frac{y^2}{16} \end{array}[/tex]

now, we want to get the area enclosed by both, and thus we'd need their points of intersection, setting both to [tex]\sqrt{y}=\cfrac{y}{4}[/tex] which in short gives us the bounds of 0 and 16.

[tex]\pi \displaystyle\int_{0}^{16}\left( y - \cfrac{y^2}{16} \right)dy\implies \pi \int_{0}^{16}y\cdot dy-\pi \int_{0}^{16}\cfrac{y^2}{16}\cdot dy\\\\\\\pi \cdot \left. \cfrac{y^2}{2} \right]_{0}^{16}-\pi \cdot \left. \cfrac{y^3}{48} \right]_{0}^{16}\implies 128\pi -\cfrac{256\pi }{3}\implies \boxed{\cfrac{128\pi }{3}}[/tex]

Find The Volume V Of The Solid Obtained By Rotating The Region Bounded By The Given Curves About The

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PLS HELP PLEASSSSS IM SERIOUSLY CRYING RN

Answers

i really don’t know 555

[tex]x = 23[/tex]

Step-by-step explanation:

First, we need to write the equations in their standard form, i.e., all the variables and their coefficients are to be placed on the left hand side and the plain numbers on the right hand side. The first equation is already in its standard form:

[tex]2x + 5y = 35\;\;\;\;\;(1)[/tex]

but we need to rewrite the 2nd one. So the standard form of the 2nd equation can be written as

[tex]x + 15y = -10\;\;\;\;\;(2)[/tex]

Now that the equations are in their standard forms, the next step that we need to do is to eliminate one of the variables, namely the y. To do that, we need to multiply either one of the equations so that when we add them together, the y-variable gets eliminated. Note if we multiply Eqn(1) by -3, we'll get -15y on one equation and a +15y on the other equation so that when they are added together, they cancel out. So if we do that, we get

[tex]-6x - 15y = -105\;\;\;\;\;(3)[/tex]

Adding Eqn(3) to Eqn(2), we get

[tex]-5x = -115[/tex]

[tex]x = \dfrac{-115}{-5} = 23[/tex]

help help help meeee

Answers

Answer:

122

Step-by-step explanation:

since line m and line n are parallel the sum of the angles is equal to 180 degrees

58 + x = 180 subtract 58 from both sides

x = 122

A firm has the following fixed proportion production function q = min(k, 2l). a. Find the firms long-run total, average, and marginal cost functions. b. Suppose that k is fixed at 20 in the short run. Calculate the firms short-run total, average, and marginal cost functions. c. Suppose v = 1 and w = 3. Calculate this firms long-run and short-run average and marginal cost curves. Q4. A firm’s production function is given by q = 2z1 + 3z2. Given the input prices, w1 for z1 and w2 for z2. a. Find the input demands z1 and z2 minimizing the cost of producing the level of output q. b. Find the cost function for producing the level q.

Answers

Answer:

I think its right

Step-by-step explanation:

a) In the long run, we have, 5k=10 and k=2l Thus, C=2l+3l= 5l=0.5q Thus, AC=5l/10l =0.5 MC=0.5 b) In the short run, k=10 Q=min(50,10l) If l<5, q=10l. C=10+3l= 10+0.3q Thus, AC=10/q + 0.3 If l>5, q=50. C= 10+3l AC= (10+3l)/50 If q>50, then MC...

Answer them all please

Answers

Answer:

see the attachment

i hope it helps u ✌️✌️✌️☠️

a) The highest common factor and the least common multiple are 12 and 48, respectively.

b) The highest common factor and the least common multiple are 12 and 72, respectively.

c) The highest common factor and the least common multiple are 8 and 160, respectively.

d) The highest common factor and the least common multiple are 12 and 48, respectively.

e) The highest common factor and the least common multiple are 14 and 840, respectively.

Procedure - Determination of the Highest Common Factor and the Least Common Multiple of two integers

In this question we must determine the highest common factor and the least common multiple of two integers.

The highest common factor consists in the maximum product of common prime numbers, of the two numbers that divides it, and the least common multiple consists in the product of prime numbers, common and not, of the two numbers, that creates a product that it is greater or equal to the greater integer.

Now, we proceed to determine the numbers for each case:

a) 12 and 48