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2 years ago

11

Is the relationship shown by the data linear? If so, model the data with an equation.

1 answer:

slega [8]2 years ago

4 0

Which ones are the x and which ones are the y

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A piece of paper is to display ~128~ 128 space, 128, space square inches of text. If there are to be one-inch margins on both si

Grace [21]

Answer:

The dimensions of the smallest piece that can be used are: 10 by 20 and the area is 200 square inches

Step-by-step explanation:

We have that:

$Area = 128$

Let the dimension of the paper be x and y;

Such that:

$Length = x$

$Width = y$

So:

$Area = x * y$

Substitute 128 for Area

$128 = x * y$

Make x the subject

$x = \frac{128}{y}$

When 1 inch margin is at top and bottom

The length becomes:

$Length = x + 1 + 1$

$Length = x + 2$

When 2 inch margin is at both sides

The width becomes:

$Width = y + 2 + 2$

$Width = y + 4$

The New Area (A) is then calculated as:

$A = (x + 2) * (y + 4)$

Substitute $\frac{128}{y}$ for x

$A = (\frac{128}{y} + 2) * (y + 4)$

Open Brackets

$A = 128 + \frac{512}{y} + 2y + 8$

Collect Like Terms

$A = \frac{512}{y} + 2y + 8+128$

$A = \frac{512}{y} + 2y + 136$

$A= 512y^{-1} + 2y + 136$

To calculate the smallest possible value of y, we have to apply calculus.

Different A with respect to y

$A' = -512y^{-2} + 2$

Set

$A' = 0$

This gives:

$0 = -512y^{-2} + 2$

Collect Like Terms

$512y^{-2} = 2$

Multiply through by $y^2$

$y^2 * 512y^{-2} = 2 * y^2$

$512 = 2y^2$

Divide through by 2

$256=y^2$

Take square roots of both sides

$\sqrt{256=y^2$

$16=y$

$y = 16$

Recall that:

$x = \frac{128}{y}$

$x = \frac{128}{16}$

$x = 8$

Recall that the new dimensions are:

$Length = x + 2$

$Width = y + 4$

So:

$Length = 8 + 2$

$Length = 10$

$Width = 16 + 4$

$Width = 20$

To double-check;

Differentiate A'

$A' = -512y^{-2} + 2$

$A" = -2 * -512y^{-3}$

$A" = 1024y^{-3}$

$A" = \frac{1024}{y^3}$

The above value is:

$A" = \frac{1024}{y^3} > 0$

This means that the calculated values are at minimum.

<em>Hence, the dimensions of the smallest piece that can be used are: 10 by 20 and the area is 200 square inches</em>

3 0

2 years ago

The height, h, of a falling object t seconds after it is dropped from a platform 300 feet above the ground is modeled by the fun

notka56 [123]

Average rate = [h(3) - h(0)]/(3 - 0)
h(3) = 300 - 16(3)^2 = 300 - 16(9) = 300 - 144 = 156
h(0) = 300 - 16(0)^2 = 300

average rate = (156 - 300)/3 = -144/3 = -48

Therefore, the object falls with an average rate of 48 ft/s during the first 3 seconds.

4 0

3 years ago

If f(x) - 4x2-3 and g(x) = x+1, find (f - g)(x).

Harrizon [31]

Answer:

4x^2-x-4

Step-by-step explanation:

f(x) = 4x^2-3 and g(x) = x+1,

(f - g)(x)= 4x^2-3 -( x+1)

Distribute the minus sign

=4x^2-3 - x-1

Combine like terms

=4x^2-x-4

6 0

2 years ago

Read 2 more answers

Members of the chess club held a bake sale to raise money. Cupcakes and cookies were sold. Cupcakes were sold for two dollars ea

Gala2k [10]

C 75.00- 1/3 were cookies which is 150 cookies. 150x0.50=75.00

5 0

2 years ago

Find all solutions to the equation in the interval [0, 2pi). COS X = sin 2x

Lera25 [3.4K]

Answer:

Step-by-step explanation:

cos x=sin 2x

sin 2x-cos x=0

2 sin x cos x-cos x=0

cos x(2 sin x-1)=0

either cos x=0

$x=\frac{\pi }{2},\frac{3\pi }{2}$

or

2 sin x-1=0

$sin x=\frac{1}{2} =sin (\frac{\pi }{6}),sin(\pi -\frac{\pi }{6} )\\x=\frac{\pi }{6} ,\frac{5 \pi }{6}$

6 0

2 years ago