A 3 inch thick slice is cut off the top of a cube, resulting in a rectangular box that has volume 128in^3 .Round your answer to two decimal places
What is the question?
x*x*(x-3) = 128
x^3 - 3 x^2 = 128
x^3 - 3 x^2 - 128 = 0
try x = 6
216 - 108 - 128 = -20
try x = 6.5
19.875
hmm try 6.25
-1.05
so try 6.20
-4.99
try 6.3
2.98
try 6.26
-.248
try 6.27
.553
so x = about 6.26
To find the volume of the original cube, we can use the formula V = s^3, where V is the volume and s is the side length of the cube.
Let's assume the side length of the original cube is x inches.
Given that a 3-inch thick slice is cut off the top of the cube, the height of the resulting rectangular box is x - 3 inches.
The volume of the rectangular box is given as 128 cubic inches.
Using the formula for the volume of a rectangular box, V = lwh, where l is the length, w is the width, and h is the height, we can form the equation
128 = x * w * (x - 3).
Since we are tasked to find the length, width, and height of the rectangular box separately, we need to find two equations.
First, we can solve for the length and width ratio using the equation:
lw = 128/(x - 3).
To find separate equations for length and width, we can split lw into two factors such that the sum of those factors gives us the volume divided by the height:
l * w = (128/(x - 3)) * (x - 3).
To split this into two factors, we can let one factor be a constant, k, and the other factor be the remaining expression:
l = k,
w = (128/(x - 3)).
We can substitute w = (128/(x - 3)) into l * w = (128/(x - 3)) * (x - 3):
l * (128/(x - 3)) = (128/(x - 3)) * (x - 3).
Simplifying, we get:
l = 128/(x - 3).
Now let's find a separate equation for the width:
w = (128/(x - 3)).
To solve for x, we can equate the length and width equations:
128/(x - 3) = (128/(x - 3)).
Simplifying, we see that the equation holds true for any value of x since both sides are equal.
Therefore, the length and width of the rectangular box can be any values, as long as the ratio of length to width is equal to 128/(x - 3).
In summary, we cannot determine the exact length and width of the rectangular box with the given information.
To solve this problem, we can use the formula for the volume of a rectangular box:
Volume = Length x Width x Height
Let's assume that the side length of the original cube is "x" inches.
Since a 3-inch thick slice is cut off from the top of the cube, the height of the rectangular box will be (x - 3) inches.
The volume of the rectangular box is given as 128 cubic inches.
So, we can write the equation as:
128 = x * Width * (x - 3)
To find the width, we need to solve this equation.
Let's simplify the equation:
128 = x(x - 3) * Width
Divide both sides of the equation by (x(x - 3)):
128 / (x(x - 3)) = Width
Now, we have the width in terms of x. To find the value of x, we can solve the equation.
Let's plug in some values for Width and calculate the value of x.
For example, if we assume the width to be 2 inches:
128 / (x(x - 3)) = 2
Multiply both sides of the equation by x(x - 3):
2x(x - 3) = 128
Expand the equation:
2x^2 - 6x = 128
Re-arrange the equation to a quadratic form:
2x^2 - 6x - 128 = 0
To solve this quadratic equation, we can use the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
For our equation, a = 2, b = -6, and c = -128.
Let's plug these values into the quadratic formula:
x = (-(-6) ± √((-6)^2 - 4 * 2 * -128)) / (2 * 2)
x = (6 ± √(36 + 1024)) / 4
x = (6 ± √1060) /4
Now, we will calculate the square root of 1060:
√1060 ≈ 32.62
Now, substitute this value into the equation:
x ≈ (6 ± 32.62) / 4
This gives us two possible solutions for x:
x1 ≈ (6 + 32.62) / 4 = 38.62 / 4 ≈ 9.65
x2 ≈ (6 - 32.62) / 4 = -26.62 / 4 ≈ -6.66
Since the side length of the cube cannot be negative, we discard x2 = -6.66.
So, the side length of the cube, x ≈ 9.65 inches.
To find the width, substitute the value of x into the equation we obtained earlier:
Width = 128 / (x(x - 3))
Width ≈ 128 / (9.65 * 6.65) ≈ 2.07 inches
Therefore, rounding our answer to two decimal places, the width of the rectangular box is approximately 2.07 inches.