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Sedbober [7]
2 years ago
7

Arelli uses 8.4 pints of blue paint and white paint to paint her bedroom walls. 3 4 of this amount is blue paint, and the rest i

s white paint. How many pints of white paint did she use to paint her bedroom walls?
Mathematics
1 answer:
Rina8888 [55]2 years ago
5 0

Answer: 2.1 Pints

Step-by-step explanation:

Given

Areli uses 8.4 Pints of blue and White Paint

If the three-fourth is blue paint i.e.

\Rightarrow 8.4\times \dfrac{3}{4}=6.3\ \text{Pints}

The remaining one-fourth is White Paint i.e.

\Rightarrow 8.4\times \dfrac{1}{4}=2.1\ \text{Pints}

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Answer:

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Step-by-step explanation:

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Solve only if you know the solution and show work.
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For the remaining integral, let t=\tan\dfrac x2. Then

\sin x=\sin\left(2\times\dfrac x2\right)=2\sin\dfrac x2\cos\dfrac x2=\dfrac{2t}{1+t^2}
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and

\mathrm dt=\dfrac12\sec^2\dfrac x2\,\mathrm dx\implies \mathrm dx=2\cos^2\dfrac x2\,\mathrm dt=\dfrac2{1+t^2}\,\mathrm dt

Now the integral is

\displaystyle\int\mathrm dx+2\int\frac{\dfrac{2t}{1+t^2}+3}{\dfrac{1-t^2}{1+t^2}+\dfrac{2t}{1+t^2}+1}\times\frac2{1+t^2}\,\mathrm dt

The first integral is trivial, so we'll focus on the latter one. You have

\displaystyle2\int\frac{2t+3(1+t^2)}{(1-t^2+2t+1+t^2)(1+t^2)}\,\mathrm dt=2\int\frac{3t^2+2t+3}{(1+t)(1+t^2)}\,\mathrm dt

Decompose the integrand into partial fractions:

\dfrac{3t^2+2t+3}{(1+t)(1+t^2)}=\dfrac2{1+t}+\dfrac{1+t}{1+t^2}

so you have

\displaystyle2\int\frac{3t^2+2t+3}{(1+t)(1+t^2)}\,\mathrm dt=4\int\frac{\mathrm dt}{1+t}+2\int\frac{\mathrm dt}{1+t^2}+\int\frac{2t}{1+t^2}\,\mathrm dt

which are all standard integrals. You end up with

\displaystyle\int\mathrm dx+4\int\frac{\mathrm dt}{1+t}+2\int\frac{\mathrm dt}{1+t^2}+\int\frac{2t}{1+t^2}\,\mathrm dt
=x+4\ln|1+t|+2\arctan t+\ln(1+t^2)+C
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To try to get the terms to match up with the available answers, let's add and subtract \ln\left|1+\tan\dfrac x2\right| to get

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which suggests A may be the answer. To make sure this is the case, show that

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You have

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Step-by-step explanation:

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