Math

Nathan is designing a box to keep his pet newt in. To make the box, he’s going to start with a solid rectangle and cut squares with sides x cm in length from each corner, as shown in the picture below. The dimensions of the of the solid rectangle are 51 cm by 45 cm. The volume of the box is 7750 cm^3

What is the length of a side of the square that Nathan is going to cut from the corners of the rectangle?

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  1. The dimensions of the box :

    51 - 2 x

    and

    45 - 2 x

    The volume of the box:

    ( 51 - 2 x ) * ( 45 - 2 x ) * x = 7750

    51 * 45 - 2 x * 45 - 2 x * 51 - 2 x * ( - 2 x ) * x = 7750

    ( 2295 - 90 x - 102 x + 4 x ^ 2 ) * x = 7750

    ( 4 x ^ 2 - 192 x + 2295 ) * x = 7750

    4 x ^ 3 - 192 x ^ 2 + 2295 x = 7750 Subtract 7750 to both sides

    4 x ^ 3 - 192 x ^ 2 + 2295 x - 7750 = 7750 - 7750

    4 x ^ 3 - 192 x ^ 2 + 2295 x - 7750 = 0

    Factorisation :

    4 x ^ 3 - 192 x ^ 2 + 2295 x - 7750 = 0

    4 x ^ 3 - 152 x ^ 2 - 40 x ^ 2 + 775 x + 1520 x - 7750 = 0

    ( 4 x ^ 3 - 152 x ^ 2 + 775 x ) - 40 x ^ 2 + 1520 x - 7750 = 0

    x ( 4 x ^ 2 - 152 x + 775 ) - 40 x ^ 2 + 1520 x - 7750 = 0

    x ( 4 x ^ 2 - 152 x + 775 ) + ( - 40 x ^ 2 + 1520 x - 7750 ) = 0

    x ( 4 x ^ 2 - 152 x + 775 ) + 10 * ( - 4x ^ 2 + 152 x - 775 ) = 0

    x ( 4 x ^ 2 - 152 x + 775 ) + 10 * ( - 1 ) ( 4x ^ 2 - 152 x + 775 ) = 0

    x ( 4 x ^ 2 - 152 x + 775 ) - 10 ( 4x ^ 2 - 152 x + 775 ) = 0

    ( 4 x ^ 2 - 152 x + 775 ) ( x - 10 ) = 0

    ( x - 10 ) ( 4x ^ 2 - 152 x + 775 ) = 0

    First solution :

    x - 10 = 0 Add 10 to both sides

    x - 10 + 10 = 0 + 10

    x = 10

    and

    4 x ^ 2 - 152 x + 775 = 0

    Other two soutions are:

    x = 19 + square root ( 669 ) / 2 = 31.93251716

    approx. x = 31.9

    and

    x = 19 - square root ( 669 ) / 2 = 6.06748284

    approx. x = 6.1

    Because the dimensions of the solid rectangle are 51 cm and 45 cm, 31.9 cannot be a solution because
    subtracting 31.9 from either of these would give you a negative number.

    51 - 2 x = 51 - 2 * 31.9 = 51 - 63.8 = - 12.8

    and

    45 - 2 x = 45 - 2 * 31.9 = 45 - 63.8 = - 58.8

    Nathan can cut out squares with that have sides of either

    6.1 cm

    OR

    10 cm

    P.S.

    If you don't know how to solve equation:

    4 x ^ 2 - 152 x + 775 = 0

    In google type :

    calculator tutorvista

    When you see list of results click on :

    Equation Calculator

    When page be open in rectangle type:

    4 x ^ 2 - 152 x + 775 = 0

    and click option :

    Calculate

    You will see solution step - by - step.

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