Question

The volume of a box VV, varies with some variable xx as V(x)=x^3 - 12x ^2 + 44x -48V(x)=x
3
−12x
2
+44x−48 cubic metres. If (x - a)(x−a) metre is the measurement of one side of the box, then find the value for aa.

Answer 1

SOLUTION

GIVEN

• The volume of a box V, varies with some variable x as V(x) = x³ - 12x² + 44x - 48 cubic metres.

• (x - a) metre is the measurement of one side of the box

TO DETERMINE

The value for a.

EVALUATION

Here it is given that the volume of a box V, varies with some variable x as

V(x) = x³ - 12x² + 44x - 48

Since (x - a) metre is the measurement of one side of the box

Thus we have

V(a) = 0

$\displaystyle \sf{ \implies {a}^{3} - 12 {a}^{2} + 44a - 48 = 0 }$

$\displaystyle \sf{ \implies {a}^{3} -2 {a}^{2} - 10 {a}^{2} + 20a + 24a - 48 = 0 }$

$\displaystyle \sf{ \implies {a}^{2}(a -2 ) - 10a(a - 2) + 24(a - 2) = 0 }$

$\displaystyle \sf{ \implies (a -2 )( {a}^{2} - 10a + 24) = 0 }$

$\displaystyle \sf{ \implies (a -2 )[ {a}^{2} -( 4 + 6)a + 24] = 0 }$

$\displaystyle \sf{ \implies (a -2 )[ {a}^{2} - 4 a - 6a + 24] = 0 }$

$\displaystyle \sf{ \implies (a -2 )[a(a - 4) - 6(a - 4)] = 0 }$

$\displaystyle \sf{ \implies (a -2 )(a - 4) (a - 6)= 0 }$

$\displaystyle \sf{ \implies a = 2 \:, \: 4 \: ,\: 6 }$

FINAL ANSWER

Hence the required value of a = 2 , 4 , 6

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