The rectangular poster shown at the right measures 2w-30. The poster has an area of 5400 cm squared. What is the value of w?

Mathematics

1 answer:

Rama09 [41]3 years ago

4 0

Complete Question:

The rectangular poster shown measures 2w-30 by w. The poster has an area of 5400 cm2. What is the value of w?

Answer:

The value of w is 60

Solution:

Given that,

The rectangular poster shown measures 2w-30 by w

Area = 5400 square centimeter

The area of rectangle is given as:

$Area = length \times width\\\\5400 = (2w - 30) \times w\\\\5400 = 2w^2 - 30w\\\\2w^2 - 30w - 5400 = 0$

Solve by quadratic formula

$\mathrm{For\:a\:quadratic\:equation\:of\:the\:form\:}ax^2+bx+c=0\mathrm{\:the\:solutions\:are\:}\\\\x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$

$\mathrm{For\:}\quad a=2,\:b=-30,\:c=-5400$

$w = \frac{-\left(-30\right)\pm \sqrt{\left(-30\right)^2-4\cdot \:2\left(-5400\right)}}{2\cdot \:2}$

$w = \frac{ 30 \pm \sqrt{44100}}{4}\\\\w = \frac{ 30 \pm 210 }{4}$

We have two solutions

$w = \frac{30+210}{4}\\\\w = 60\\\\And\\\\w = \frac{30-210}{4}\\\\w=-45$

Ignore, negative value as width cannot be negative

Thus value of w is 60

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Express this to single logarithm

zzz [600]

Answer: $\log_{2}\left(\frac{q^2\sqrt{m}}{n^3}\right)$

We have something in the form log(x/y) where x = q^2*sqrt(m) and y = n^3. The log is base 2.

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Explanation:

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I'm going to use these three log rules, which apply to any base.

log(A) + log(B) = log(A*B)
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From there, we can then say the following:

$\frac{1}{2}\log_{2}\left(m\right)-3\log_{2}\left(n\right)+2\log_{2}\left(q\right)\\\\\log_{2}\left(m^{1/2}\right)-\log_{2}\left(n^3\right)+\log_{2}\left(q^2\right) \ \text{ .... use log rule 3}\\\\\log_{2}\left(\sqrt{m}\right)+\log_{2}\left(q^2\right)-\log_{2}\left(n^3\right)\\\\\log_{2}\left(\sqrt{m}*q^2\right)-\log_{2}\left(n^3\right) \ \text{ .... use log rule 1}\\\\\log_{2}\left(\frac{q^2\sqrt{m}}{n^3}\right) \ \text{ .... use log rule 2}$