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3 years ago

What is the factorization of the polynomial below?

Mathematics

2 answers:

tia_tia [17]3 years ago

3 0

I'm going to go with D

adoni [48]3 years ago

3 0

Has to be D

Explanation:
3x²+36x+81
3(x²+12x+27)
3(x²+3x+9x+27)
3{x(x+3)+9(x+3)}
3(x+3)(x+9)

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A Norman window is a rectangle with a semicircle on top. Suppose that the perimeter of a particular Norman window is to be 25 fe

Varvara68 [4.7K]

Answer:

$Length =\frac{25}{4 + \pi}$ and $Width = \frac{50}{4+\pi}$

Step-by-step explanation:

This question is better understood with an attachment.

See attachment for illustration.

Given

Represent Perimeter with P

$P = 25ft$

Required

Determine the dimension of the rectangle that maximizes the area

First, we calculate the perimeter of the rectangular part of the window.

From the attachment, the rectangle is not closed at the top.

So, The perimeter would be the sum of the three closed sides

Where

$Width = 2x$

$Length = y$

So:

$P_{Rectangle} = y + y + 2x$

$P_{Rectangle} = 2y + 2x$

Next, we determine the circumference of the semi circle.

Circumference of a semicircle is calculated as:

$C = \frac{1}{2}\pi r$

From the attachment,

$Radius (r) = x$

So, we have:

$C = \frac{1}{2}2\pi * x$

$C = \pi x$

So, the perimeter of the window is:

$P = P_{Rectangle} + C$

$P =2y + 2x + \pi x$

Recall that: $P = 25$

So, we have:

$25 =2y + 2x +\pi x$

Make 2y the subject

$2y = 25 - 2x - \pi x$

Make y the subject:

$y = \frac{25}{2} - \frac{2x}{2} - \frac{\pi x}{2}$

$y = \frac{25}{2} - x - \frac{\pi x}{2}$

Next, we determine the area (A) of the window

A = Area of Rectangle + Area of Semicircle

$A = 2x * y + \frac{1}{2}\pi r^2$

$A = 2xy + \frac{1}{2}\pi r^2$

Recall that

$Radius (r) = x$

$A = 2xy + \frac{1}{2}\pi x^2$

Substitute $\frac{25}{2} - x - \frac{\pi x}{2}$ for y in $A = 2xy + \frac{1}{2}\pi x^2$

$A = 2x(\frac{25}{2} - x - \frac{\pi x}{2}) + \frac{1}{2}\pi x^2$

Open Bracket

$A = 2x * \frac{25}{2} - 2x * x - 2x * \frac{\pi x}{2} + \frac{1}{2}\pi x^2$

$A = 25x - 2x^2 - \pi x^2 + \frac{1}{2}\pi x^2$

$A = 25x - 2x^2 - \frac{1}{2}\pi x^2$

To maximize area, we have to determine differentiate both sides and set A' = 0

Differentiate

$A' = 25 - 4x - \pi x$

$A' = 0$

So, we have:

$0 = 25 - 4x - \pi x$

Factorize:

$0 = 25 -x(4 + \pi)$

$-25 =-x(4 + \pi)$

Solve for x

$x = \frac{-25}{-(4+\pi)}$

$x = \frac{25}{4+\pi}$

Recall that

$Width = 2x$

$Width = 2(\frac{25}{4+\pi})$

$Width = \frac{50}{4+\pi}$

Recall that:

$y = \frac{25}{2} - x - \frac{\pi x}{2}$

Substitute $\frac{25}{4+\pi}$ for x

$y = \frac{25}{2} - (\frac{25}{4+\pi}) - \frac{\pi (\frac{25}{4+\pi})}{2}$

$y = \frac{25}{2} - (\frac{25}{4+\pi}) - \frac{\frac{25\pi}{4+\pi}}{2}$

$y = \frac{25}{2} - (\frac{25}{4+\pi}) - \frac{25\pi}{4+\pi} * \frac{1}{2}$

$y = \frac{25}{2} - \frac{25}{4+\pi} - \frac{25\pi}{2(4+\pi)}$

$y = \frac{25(4+\pi) - 25 * 2 - 25\pi}{2(4 + \pi)}$

$y = \frac{100+25\pi - 50 - 25\pi}{2(4 + \pi)}$

$y = \frac{100- 50+25\pi - 25\pi}{2(4 + \pi)}$

$y = \frac{50}{2(4 + \pi)}$

$y = \frac{25}{4 + \pi}$

Recall that:

$Length = y$

So:

$Length =\frac{25}{4 + \pi}$

Hence, the dimension of the rectangle is:

$Length =\frac{25}{4 + \pi}$ and $Width = \frac{50}{4+\pi}$

3 0

3 years ago

When a figure is translated on a coordinate grid, what conclusion can you draw from the pre-image and image? (2 points) Group of

Nataly [62]

Answer:

B.the size of the figures change

Step-by-step explanation:

8 0

4 years ago

Find the equation of the line passing through the given points (-5,6) and (-5,8)

diamong [38]

Answer:

x = -5

Step-by-step explanation:

To find the equation of a line, use the formula y = mx + b, where m is the slope of the line and b is the y intercept.

To find the slope, use your ordered pairs: m = (y2 - y1)/(x2 - x1). Substituting in your values, m = (8-6)/(-5 - -5), or m = 2/0. Since you cannot divide by 0, the slope is undefined; you have a vertical line.

If you plot the points on a graph, (-5, 6) and (-5, 8) both pass through -5 on the x axis. Since this is a vertical line, it will never cross the y axis, so you will not have a y intercept.

Your equation is x = -5.

8 0

4 years ago

73 = 87 - n what is n

enot [183]

Answer: 14

Step-by-step explanation:

Make n the subject.

73 = 87 - n

n + 73 = 87

n = 87 - 73

n = 14

6 0

3 years ago

Read 2 more answers

Given four functions, which one will have the smallest y-intercept?

Natalka [10]

F(x) function has a starting value of $6.00 and increases from there.So the y-intercept is 6.

g(x) function shows in the table that when x = 0, y = 6. This is the y-intercept.

h(x) shows on the graph that the y-intercept is 4.

j(x) is an exponential growth function, a * b^x where a is the y-intercept.
y = 10^ x^x has a y-intercept at 10.

The smallest y-intercept is function h(x).

3 0

4 years ago