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zhannawk [14.2K]

3 years ago

12

A rectangular photo frame covers an area of 20 square inches on a wall. The width of the frame is 1 inch less than its length. F

ind the length and width of the photo frame.

1 answer:

Salsk061 [2.6K]3 years ago

7 0

You need to find two factors of 20 that are consecutive: 4,5.
Answer: 5inx4in

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1. (5pts) Find the derivatives of the function using the definition of derivative.

andreyandreev [35.5K]

2.8.1

$f(x) = \dfrac4{\sqrt{3-x}}$

By definition of the derivative,

$f'(x) = \displaystyle \lim_{h\to0} \frac{f(x+h)-f(x)}{h}$

We have

$f(x+h) = \dfrac4{\sqrt{3-(x+h)}}$

and

$f(x+h)-f(x) = \dfrac4{\sqrt{3-(x+h)}} - \dfrac4{\sqrt{3-x}}$

Combine these fractions into one with a common denominator:

$f(x+h)-f(x) = \dfrac{4\sqrt{3-x} - 4\sqrt{3-(x+h)}}{\sqrt{3-x}\sqrt{3-(x+h)}}$

Rationalize the numerator by multiplying uniformly by the conjugate of the numerator, and simplify the result:

$f(x+h) - f(x) = \dfrac{\left(4\sqrt{3-x} - 4\sqrt{3-(x+h)}\right)\left(4\sqrt{3-x} + 4\sqrt{3-(x+h)}\right)}{\sqrt{3-x}\sqrt{3-(x+h)}\left(4\sqrt{3-x} + 4\sqrt{3-(x+h)}\right)} \\\\ f(x+h) - f(x) = \dfrac{\left(4\sqrt{3-x}\right)^2 - \left(4\sqrt{3-(x+h)}\right)^2}{\sqrt{3-x}\sqrt{3-(x+h)}\left(4\sqrt{3-x} + 4\sqrt{3-(x+h)}\right)} \\\\ f(x+h) - f(x) = \dfrac{16(3-x) - 16(3-(x+h))}{\sqrt{3-x}\sqrt{3-(x+h)}\left(4\sqrt{3-x} + 4\sqrt{3-(x+h)}\right)} \\\\ f(x+h) - f(x) = \dfrac{16h}{\sqrt{3-x}\sqrt{3-(x+h)}\left(4\sqrt{3-x} + 4\sqrt{3-(x+h)}\right)}$

Now divide this by <em>h</em> and take the limit as <em>h</em> approaches 0 :

$\dfrac{f(x+h)-f(x)}h = \dfrac{16}{\sqrt{3-x}\sqrt{3-(x+h)}\left(4\sqrt{3-x} + 4\sqrt{3-(x+h)}\right)} \\\\ \displaystyle \lim_{h\to0}\frac{f(x+h)-f(x)}h = \dfrac{16}{\sqrt{3-x}\sqrt{3-x}\left(4\sqrt{3-x} + 4\sqrt{3-x}\right)} \\\\ \implies f'(x) = \dfrac{16}{4\left(\sqrt{3-x}\right)^3} = \boxed{\dfrac4{(3-x)^{3/2}}}$

3.1.1.

$f(x) = 4x^5 - \dfrac1{4x^2} + \sqrt[3]{x} - \pi^2 + 10e^3$

Differentiate one term at a time:

• power rule

$\left(4x^5\right)' = 4\left(x^5\right)' = 4\cdot5x^4 = 20x^4$

$\left(\dfrac1{4x^2}\right)' = \dfrac14\left(x^{-2}\right)' = \dfrac14\cdot-2x^{-3} = -\dfrac1{2x^3}$

$\left(\sqrt[3]{x}\right)' = \left(x^{1/3}\right)' = \dfrac13 x^{-2/3} = \dfrac1{3x^{2/3}}$

The last two terms are constant, so their derivatives are both zero.

So you end up with

$f'(x) = \boxed{20x^4 + \dfrac1{2x^3} + \dfrac1{3x^{2/3}}}$

8 0

2 years ago

A factory makes 1,200 shirts every 6 hours. The factory makes shirts for 9 hours each workday. Enter the fewest number of workda

eduard

<span>The factory makes shirts for 9 hours each workday. </span>

7 0

3 years ago

Sa = 2(pi)(r)(h) + 2(pi)(r^2) v = (pi)(r^2)(h) find the surface area and the volume.

LenKa [72]

If Sa=2πrh+2π $r^{2}$ v=π $r^{2}h$ then the surface area is π $r^{2} h$ and volume is

(rh-2h)/2r.

Given Sa=2πrh+2π $r^{2} v$ =π $r^{2}h$ .

We have to find surface area and volume from the given expression.

Volume is basically amount of substance a container can hold in its capacity.

First we will find the value of v from the expression. Because they are in equal to each other, we can easily find the value of v.

2πrh+2π $r^{2}$ v=π $r^{2}$ h

Keeping the term containing v at left side and take all other to right side.

2π $r^{2}$ v=π $r^{2}h$ -2πrh

v=(π $r^{2}$ h-2πrh)/2π $r^{2}$

v=π $r^{2} h$ /2π $r^{2}$ -2πrh/2π $r^{2}$

v=h/2-h/r

v=h(r-2)/2r

Put the value of v in Sa=2πrh+2π $r^{2} v$

Sa=2πrh+2π $r^{2}$ *h(r-2)/2r

=2πrh+2πrh(r-2)/2

=2πrh+πrh(r-2)

=2πrh+π $r^{2}$ h-2πrh

=π $r^{2}$ h

Hence surface area is π $r^{2}$ h and volume is h(r-2)/2.

Learn more about surface area at brainly.com/question/16519513

#SPJ4

6 0

2 years ago

Write and expression equal to 7x8

Lubov Fominskaja [6]

(4+3)x(4+4) this should be the answer

3 0

3 years ago

Read 2 more answers

2.) If ZA = (20 – 2m)º and ZD = (m – 130°, what is the value of m?

vivado [14]

Answer:

m = 50

Step-by-step explanation:

∠ D and ∠ A are vertically opposite angles and are congruent , then

m - 130 = 20 - 2m ( add 2m to both sides )

3m - 130 = 20 ( add 130 to both sides )

3m = 150 ( divide both sides by 3 )

m = 50

3 0

2 years ago

Read 2 more answers