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3 years ago

To the nearest millimeter, a cell phone is 82 mm long and 36 mm wide. What is the ratio of the width to the length?

Mathematics

1 answer:

iVinArrow [24]3 years ago

3 0

<h2>The ratio of the width to the length of a cell phone is 18 : 41.</h2>

Step-by-step explanation:

Given,

The length of a cell phone(l) = 82 mm and

The width of a cell phone(b) = 36 mm

Find, the ratio of the width to the length of a cell phone = ?

∴ The ratio of the width to the length of a cell phone

= 36 mm : 82 mm

= 36 : 82

Divided by 2, we get

= $\dfrac{36}{2}$ : $\dfrac{82}{2}$

= 18 : 41

Thus, the ratio of the width to the length of a cell phone is 18 : 41.

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The formula for the length of arc of the function f(x) in the interval [a,b] is:

$\displaystyle\int_a^b \sqrt{1+[f'(x)]^2}dx$

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$f'(x)=\frac{1}{x}$

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$\displaystyle\int_a^b g(x)}dx\approx\frac{\Delta x}{3}\left( g(a)+4g(a+\Delta x)+2g(a+2\Delta x) +4g(a+3\Delta x)+g(b) \right)$

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$a=1,b=3,n=4, \displaystyle\Delta x=\frac{b-a}{n}=\frac{2}{4}=\frac{1}{2},g(x)= \sqrt{1+\frac{1}{x^2}}$

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$\displaystyle\int_1^3\sqrt{1+\frac{1}{x^2}}dx\\\approx \frac{\frac{1}{3}}{3}\left( \sqrt{1+\frac{1}{1^2}} +4\sqrt{1+\frac{1}{\left(1+\frac{1}{2}\right)^2}} +2\sqrt{1+\frac{1}{\left(1+\frac{2}{2}\right)^2}} +4\sqrt{1+\frac{1}{\left(1+\frac{3}{2}\right)^2}} +\sqrt{1+\frac{1}{3^2}} \right)$

Then simplifying a bit:

$\displaystyle\int_1^3\sqrt{1+\frac{1}{x^2}}dx \approx \frac{1}{9}\left( \sqrt{1+\frac{1}{1^2}} +4\sqrt{1+\frac{1}{\left(\frac{3}{2}\right)^2}} +2\sqrt{1+\frac{1}{\left(2\right)^2}} +4\sqrt{1+\frac{1}{\left(\frac{5}{2}\right)^2}} +\sqrt{1+\frac{1}{3^2}} \right)$

Then we just do those computations and we finally get the approximation via Simposn's rule:

$\displaystyle\int_1^3\sqrt{1+\frac{1}{x^2}}dx\approx 1.5355453$

While when we do the integral by using technology we get: 2.3020.

The approximation with Simpon’s rule is close but about 33% smaller:

$\displaystyle\frac{2.3020-1.5355453}{2.3020}\cdot100\%\approx 33\%$

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3 years ago

To the nearest​ millimeter, a cell phone is 82 mm long and 36 mm wide. What is the ratio of the width to the​ length?

To the nearest millimeter, a cell phone is 82 mm long and 36 mm wide. What is the ratio of the width to the length?