5kg of wheat and 3kg of barley cost $424, whereas 8kg of wheat and 4kg of barley cost $628. Find the cost of 1kg of each item.
Answers
ForSphere
\rm Radius \: (r) = 4.2 \: cmRadius(r)=4.2cm
\therefore \sf Volume \: of \: sphere = \frac{4}{3}\pi r {}^{2} ∴Volumeofsphere=
3
4
πr
2
\: \: \: \: \: \: \sf = \frac{4}{3}\pi (4.2) {}^{3} \: cm {}^{3} =
3
4
π(4.2)
3
cm
3
\bf {For \: Cylinder}ForCylinder
\rm Radius(R) = 6 \: cmRadius(R)=6cm
\sf{Let \: the \: height \: of \: the \: cylinder \: be \: H \: cm.}LettheheightofthecylinderbeHcm.
\rm Then, \: Then,
\sf{Volume \: of \: Cylinder = \pi r {}^{2} H = \pi(6) {}^{2} H \: cm {}^{3} }VolumeofCylinder=πr
2
H=π(6)
2
Hcm
3
\rm According \: to \: the \: Question,AccordingtotheQuestion,
\sf{Volume \: of \: the \: metallic \: sphere \: must \: be}Volumeofthemetallicspheremustbe
\rm = Volume \: of \: the \: cylinder=Volumeofthecylinder
\implies \: \: \: \: \sf\frac{4}{3} \: \pi(4.2) {}^{3} = \pi(6) {}^{2} h⟹
3
4
π(4.2)
3
=π(6)
2
h
\rm Dividing \: both \: sides \: by \: \pi \: and \: cross \: multiplying,Dividingbothsidesbyπandcrossmultiplying,
\: \: \: \: \: \: \: \: \sf{3(6) {}^{2} H = 4(4.2) {}^{3} }3(6)
2
H=4(4.2)
3
\begin{gathered} \implies \: \: \: \: \: \: \sf{ H = \frac{4(4.2) {}^{3}} {3(6) {}^{2}}} \\ \end{gathered}
⟹H=
3(6)
2
4(4.2)
3
\begin{gathered} \implies \: \: \rm H = \frac{4 \times 4.2 \times 4.2 \times 4.2}{3 \times 6 \times 6} \\ \end{gathered}
⟹H=
3×6×6
4×4.2×4.2×4.2
\: \: \: \: \: \: \: \sf = 4 \times 1.4 \times 0.7 \times 0.7=4×1.4×0.7×0.7
\begin{gathered} \: \: \: \: \: \: \: \: \: \rm = 4 \times \frac{14}{10} \times \frac{7}{10} \times \frac{7}{10} \\ \end{gathered}
=4×
10
14
×
10
7
×
10
7
\begin{gathered} \: \: \: \: \: \: \sf = \frac{56 \times 49}{1000} = \frac{2744}{1000} \\ \end{gathered}
=
1000
56×49
=
1000
2744
\implies \rm \: \: \: \: \: \: \: \: H = 2.744⟹H=2.744
\begin{gathered} \sf Hence, \: the \: height \: of \: the \: cylinder \: is \: 2.744 \\ \: \sf \:cm.\end{gathered}
Hence,theheightofthecylinderis2.744
cm.