Mathematics: A gardener fills a 2 1/2 cubic foot wheelbarrow 2/3 of the way without soil weight 75.25 pounds. What is the weight of the soil un wheelbarrow?

★ Formula Applied :★ Explanation :Lets use substitution method ,Let , u = sin2x⇒ du = 2.cos2x.dx★ Alternate Method :]...

A gardener fills a 2 1/2 cubic foot wheelbarrow 2/3

Ответ:

robrod75

03.02.2022

Mathematics

Volume of wheelbarrow , V = 2 1/2 ft³ = 5/2 ft³.

Volume of soil in wheelbarrow , $v = \dfrac{5}{3}\ ft^3$ .

Mass of soil in wheelbarrow , m = 75.25 pounds.

Now, volume left , $v'=\dfrac{5}{6}\ ft^3$ .

Let, mass of soil un wheelbarrow is x.

$x=\dfrac{\dfrac{5}{6}\times 75.25}{\dfrac{5}{3}}\\\\x=37.625\ pounds$

Hence, this is the required solution.

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Ответ:

von2001

27.07.2021

Mathematics

★ Formula Applied :

$\begin{gathered}\sf \bullet\ \; cos^2x-sin^2x=cos2x\\\\\to\ \sf \pink{sin^2x-cos^2x=-cos2x}\end{gathered}$

$\begin{gathered}\bullet\ \; \sf sin2x=2.sinx.cosx\\\\\to \sf \blue{sinx.cosx=\dfrac{sin2x}{2}}\end{gathered}$

$\displaystyle \bullet\ \; \sf \int \dfrac{dx}{x}$

$\bullet\ \; \sf \ln (ab)=\ln a+\ln$

★ Explanation :

$\begin{gathered}\displaystyle \sf \int \dfrac{sin^2x-cos^2x}{sinx.cosx}dx\\\\\to \sf \int \dfrac{-cos2x}{\frac{sin2x}{2}}dx\\\\\to \sf \int \dfrac{-2cos2x}{sin2x}dx\end{gathered}$

Lets use substitution method ,

Let , u = sin2x

⇒ du = 2.cos2x.dx

$\begin{gathered}\to \displaystyle \sf \int \dfrac{-du}{u}\\\\\to \sf -\int \dfrac{du}{u}\\\\\to \sf -ln|u|\\\\\end{gathered}$

$\to \sf \red{-ln|sin2x|+c}$

$\to \sf - \ln |2sinx.cosx|+c$

$\to \sf - \ln |sinx.cosx|+(\ln 2+c)$

$\leadsto - \sf \red{\ln |sinx.cosx|+c}\ \; \bigstar$

★ Alternate Method :

$\displaystyle \sf \int \dfrac{sin^2x-cos^2x}{sinx.cosx}$

$\displaystyle \to \sf \int \left( \dfrac{sin^2x}{sinx.cosx}-\dfrac{cos^2x}{sinx.cosx}\right)dx$

$\begin{gathered}\displaystyle \to \sf \int \left( \dfrac{sinx}{cosx} -\dfrac{cosx}{sinx} \right)dx\\\\\to\ \sf \int (tanx-cotx)dx\\\\\to \sf \int tanx.dx-\int cotx.dx\\\\\to \sf ln|secx|-ln|sinx|+c\end{gathered}$

$\to \sf ln\left| \dfrac{secx}{sinx}\right|+c$

$\begin{gathered}\to \sf ln\left| \dfrac{1}{sinx.cosx} \right|+c\\\\\end{gathered}$ ]

$\leadsto \sf \pink{-ln|sinx.cosx|+c}\ \; \bigstar$

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A gardener fills a 2 1/2 cubic foot wheelbarrow 2/3 of the way without soil weight 75.25 pounds. What is the weight of the soil un wheelbarrow?

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