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2 years ago

8

The length of a rectangle is one inch less than three times it's width. If the perimeter of the rectangle is 54 inches, find the

length.

1 answer:

Mashcka [7]2 years ago

8 0

Answer:

l = 20

Step-by-step explanation:

set up a system of equations

l = 3w -1

54 = 2l + 2w

substitute

54 = 2(3w-1) + 2w

56 = 8w

w = 7

l = 3 *7 -1

l = 20

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2 years ago

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When temperature is zero degree Celsius, the Fahrenheit temperature is 32. When the Celsius temperature is 100, the correspondin

7nadin3 [17]

Answer:

$\\ y = 1.8(x) + 32$ or $\\ y = \frac{9}{5}(x) + 32$

or equivalently:

$\\ F = 1.8(C) + 32$ or $\\ F = \frac{9}{5}(C) + 32$

Step-by-step explanation:

To express the Fahrenheit temperature <em>as a linear function of the Celsius temperature</em>, F(c), we can proceed as follows.

We can use here <em>the two-point form</em> <em>equation</em> of a line:

$\\ y-y_1 = \frac{y_2 - y_1}{x_2 - x_1}(x-x_1)$ [1]

We are asked to express the <em>Fahrenheit temperature</em> as a function of <em>Celsius temperature</em>, so the independent variable, in this case, is <em>x</em> (Celsius temperature) and the dependent variable is <em>y</em> (Fahrenheit temperature).

When temperature is zero degree Celsius ( $\\x_1 = 0$ ), the Fahrenheit temperature is 32 ( $\\y_1 = 32$ ).

When the Celsius temperature is 100 ( $\\x_2 = 100$ ), the corresponding Fahrenheit temperature is 212 ( $\\y_2 = 212$ ).

Then, using [1], we have:

$\\ y-32 = \frac{212 - 32}{100 - 0}(x-0)$

$\\ y-32 = \frac{180}{100}(x)$

$\\ y-32 = 1.8(x)$ .

It could be also be written as:

$\\ y-32 = \frac{18}{10}(x)$ = $\\ y-32 = \frac{9}{5}(x)$ , as it commonly appears in books.

Then <em>the Fahrenheit temperature express as a linear function of the Celsius temperature, F(c</em>) is ( solving the equation for <em>y </em>) :

$\\ y = 1.8(x) + 32$ or $\\ y = \frac{9}{5}(x) + 32$ .

Or equivalently:

$\\ F = 1.8(C) + 32$ or $\\ F = \frac{9}{5}(C) + 32$

We can check this using the given values from the question:

For 0 Celsius degrees, the Fahrenheit temperature is:

$\\ y = 1.8(0) + 32$ = 32 Fahrenheit degrees.

For 100 Celsius degrees, the Fahrenheit temperature is:

$\\ y = 1.8(100) + 32 = 180 + 32$ = 212 Fahrenheit degrees.

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2 years ago

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Fynjy0 [20]

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Step-by-step explanation:

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In each case, the third-level factoring mentioned in step 4 is the factoring of the difference of squares: a² -b² = (a -b)(a +b).

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The step-by-step is exactly what you need to do. It is simply a matter of following those instructions. You do have to be able to recognize the common factors of a pair of terms. That will be the GCF of the numbers and the least powers of the common variables.

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2 years ago

How to find the derivative of cos^2x? i seem to be confused.

slamgirl [31]

If you're using the app, try seeing this answer through your browser: brainly.com/question/2927231

————————

You can actually use either the product rule or the chain rule for this one. Observe:

• Method I:

y = cos² x

y = cos x · cos x

Differentiate it by applying the product rule:

$\mathsf{\dfrac{dy}{dx}=\dfrac{d}{dx}(cos\,x\cdot cos\,x)}\\\\\\
\mathsf{\dfrac{dy}{dx}=\dfrac{d}{dx}(cos\,x)\cdot cos\,x+cos\,x\cdot \dfrac{d}{dx}(cos\,x)}$

The derivative of cos x is – sin x. So you have

$\mathsf{\dfrac{dy}{dx}=(-sin\,x)\cdot cos\,x+cos\,x\cdot (-sin\,x)}\\\\\\
\mathsf{\dfrac{dy}{dx}=-sin\,x\cdot cos\,x-cos\,x\cdot sin\,x}$

$\therefore~~\boxed{\begin{array}{c}\mathsf{\dfrac{dy}{dx}=-2\,sin\,x\cdot cos\,x}\end{array}}\qquad\quad\checkmark$

—————

• Method II:

You can also treat y as a composite function:

$\left\{\!
\begin{array}{l}
\mathsf{y=u^2}\\\\
\mathsf{u=cos\,x}
\end{array}
\right.$

and then, differentiate y by applying the chain rule:

$\mathsf{\dfrac{dy}{dx}=\dfrac{dy}{du}\cdot \dfrac{du}{dx}}\\\\\\
\mathsf{\dfrac{dy}{dx}=\dfrac{d}{du}(u^2)\cdot \dfrac{d}{dx}(cos\,x)}$

For that first derivative with respect to u, just use the power rule, then you have

$\mathsf{\dfrac{dy}{dx}=2u^{2-1}\cdot \dfrac{d}{dx}(cos\,x)}\\\\\\
\mathsf{\dfrac{dy}{dx}=2u\cdot (-sin\,x)\qquad\quad (but~~u=cos\,x)}\\\\\\
\mathsf{\dfrac{dy}{dx}=2\,cos\,x\cdot (-sin\,x)}$

and then you get the same answer:

$\therefore~~\boxed{\begin{array}{c}\mathsf{\dfrac{dy}{dx}=-2\,sin\,x\cdot cos\,x}\end{array}}\qquad\quad\checkmark$

I hope this helps. =)

Tags: <em>derivative chain rule product rule composite function trigonometric trig squared cosine cos differential integral calculus</em>

3 0

3 years ago