Answer by factoring a quadratic equation.
The length of an open-top box is 4 cm longer than its width. The box was made from a 480-cm^2 rectangular sheet of material with 6cm by 6cm squares cut from each corner. The height of the box is 6cm. Find the dimensions of the box.
Please show me in detail how to set this up and solve.
To solve this problem, we will use the information provided to set up and solve a quadratic equation.
Let's start by setting up the equation. We are given that the length of the box is 4 cm longer than the width. Let's say the width of the box is 'x' cm. Therefore, the length of the box would be 'x+4' cm.
Now, we can find the area of the base of the box, which is the product of the length and width. The area of the base is given by:
Area = Length * Width
Since we know the area is 480 cm^2, we can write the equation as:
(x + 4) * x = 480
Now, let's simplify this equation by expanding the expression:
x^2 + 4x = 480
To solve this equation, we need to rearrange it into the standard quadratic form, which is ax^2 + bx + c = 0. In this case, our quadratic equation is already in that form.
Next, we move all the terms to one side of the equation to obtain:
x^2 + 4x - 480 = 0
Now, we can solve this quadratic equation by factoring. We need to find two numbers that multiply to give 'a' (1 in this case) multiplied by 'c' (-480 in this case), and add up to give 'b' (4 in this case).
Let's factorize the quadratic equation:
x^2 + 4x - 480 = 0
(x + 24)(x - 20) = 0
Now, we set each factor equal to zero and solve for 'x':
x + 24 = 0 --OR-- x - 20 = 0
x = -24 --OR-- x = 20
Since we're talking about dimensions, the width cannot be negative. Therefore, the width of the box is 20 cm.
To find the length, we can substitute the value of the width into the equation for length:
Length = Width + 4
Length = 20 + 4
Length = 24 cm
So, the dimensions of the box are 20 cm (width) x 24 cm (length) x 6 cm (height).