Question

The base of a triangular field is three times its altitude. If the cost of levelling the field at * 30.50 per sq.
metre is 7350, find its base and height.
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Answer 1

$\bf \underline{ \underline{\maltese\:Given} }$

$\sf The \: base \: of \: a \: triangular \: field \: is \: three \: times \: its \: altitude.$

$\sf Cost \: of \: levelling \: the \: field \: at \: 30.50 \: per \: m^{2} \: is \: Rs. \: 7350$

$\bf \underline{ \underline{\maltese\:To \: find } }$

$\sf \implies Base \: and \: height = \: ?$

$\bf \underline{ \underline{\maltese \: Solution } }$

$\sf Let \: us \: assume \: that,$

$\sf \implies Height \: of \: the \: triangular \: field \: be \: h$

$\sf \implies \bf{Then}, \: \sf the \: base \: of \: the \: triangular \: field \: will \: be \: 3h$

$\bf \underline{Now},$

$\underline{ \boxed{ \sf \: Area \: of \: triangular \: field = \dfrac{1}{2} \times Base \times Height}}$

$\sf Substituting \: the \: values,$

$\sf \: Area \: of \: triangular \: field = \dfrac{1}{2} \times 3h \times h$

$\sf \: Area \: of \: triangular \: field = \dfrac{3h^{2} }{2}$

$\bf \underline{Now},$

$\sf Cost \: of \: cultivating \: per \: m^{2} \: field = Rs. \: 30.50$

$\sf Cost \: of \: cultivating \: whole \: field = Rs. \: 7350$

$\bf \underline{Then },$

$\sf Area \: of \: field = \dfrac{Total \: cost}{ Rate }$

$\sf Area \: of \: field = \dfrac{7350}{ 30.50 } = 240.98$

$\bf \underline{Therefore },$

$\sf \implies \dfrac{3h^{2} }{2} =240.98$

$\sf \implies h^{2} = 240.98 \times \dfrac{2}{3}$

$\sf \implies h^{2} = 160.65$

$\sf \implies h = \sqrt{160.65}$

$\sf \implies Thus, \: height = \bf 12.67 \: m$

$\sf \: Then, \: the \: base \: will \: be \: 12.67 \times 3$

$\sf \implies Base = 12.67 \times 3 = \bf 38.01 \: m$