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?
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.
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4 x ^ 2 - 152 x + 775 = 0
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4 x ^ 2 - 152 x + 775 = 0
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You will see solution step - by - step.
To find the side length of the square that Nathan is going to cut from the corners of the rectangle, we need to use the following formula:
Volume of the box = (length - 2x)(width - 2x)(x)
Given that the volume of the box is 7750 cm^3, the length is 51 cm, and the width is 45 cm, we can substitute these values into the formula:
7750 = (51 - 2x)(45 - 2x)(x)
Now, we need to solve this equation to find the value of x.
To find the length of a side of the square that Nathan is going to cut from the corners of the rectangle, we can use the equation for the volume of a rectangular box.
The volume of a rectangular box is given by the equation:
Volume = Length × Width × Height
In this case, the length and width of the solid rectangle are given as 51 cm and 45 cm, respectively. Let's assume that the side length of the square that Nathan is going to cut is x cm.
When Nathan cuts squares with sides x cm from each corner, the resulting dimensions of the rectangular box will be:
Length = 51 cm - 2x cm (as he cuts x cm from each side)
Width = 45 cm - 2x cm
Given that the volume of the resulting box is 7750 cm^3, we can set up an equation using the volume formula:
(51 - 2x) × (45 - 2x) × x = 7750
This is a quadratic equation. Let's solve it.
First, expand the equation:
(45 - 2x)(51 - 2x)x = 7750
Now, distribute:
(45 × 51 - 90x - 102x + 4x^2)x = 7750
Combine like terms:
(2295 - 192x + 4x^2)x = 7750
Now, simplify further:
4x^3 - 192x^2 + 2295x - 7750 = 0
This equation can be solved using factoring, the quadratic formula, or numerical methods such as graphing or using a calculator.
After solving the equation, you will find the value(s) of x, which represents the length of a side of the square that Nathan is going to cut from the corners of the rectangle.