A rectangular prism has a length of x+2, a width of x+1, and a height of 4. The volume of this rectangular prism is 24. What is the length and width of this rectangular prism.
The volume of a rectangular prism is given by the formula V = length * width * height. We are given that the volume is 24, so we can set up an equation as follows:
24 = (x+2) * (x+1) * 4
To solve this equation, we can simplify and express it in quadratic form:
24 = 4(x+2)(x+1)
Dividing both sides of the equation by 4:
6 = (x+2)(x+1)
Expanding the equation:
6 = x^2 + 3x + 2
Rearranging the equation to set it equal to zero:
x^2 + 3x + 2 - 6 = 0
Simplifying:
x^2 + 3x - 4 = 0
To factor this quadratic equation, we need to find two numbers that multiply to -4 and add up to 3. These numbers are 4 and -1:
(x + 4)(x - 1) = 0
Setting each factor equal to zero:
x + 4 = 0 or x - 1 = 0
Solving for x:
x = -4 or x = 1
Since we are looking for the dimensions of a rectangular prism, the length and width cannot be negative. Therefore, x = 1.
Let's substitute x = 1 back into the equation to find the length and width:
Length = x + 2 = 1 + 2 = 3
Width = x + 1 = 1 + 1 = 2
Therefore, the length of the rectangular prism is 3 units, and the width is 2 units.
To find the length and width of the rectangular prism, we can use the formula for the volume of a rectangular prism, which is given by:
Volume = length × width × height
In this case, we are given that the volume of the rectangular prism is 24 and the height is 4. We need to find the values of length and width.
Plugging in the given values, we get:
24 = (x+2) × (x+1) × 4
To solve this equation, let's simplify it step by step.
Step 1: Distribute the 4 to the terms inside the parentheses:
24 = 4(x^2 + 3x + 2)
Step 2: Simplify further:
24 = 4x^2 + 12x + 8
Step 3: Divide both sides of the equation by 4 to isolate the quadratic expression:
6 = x^2 + 3x + 2
Step 4: Rearrange the equation by subtracting 6 from both sides:
x^2 + 3x + 2 - 6 = 0
x^2 + 3x - 4 = 0
Step 5: Factor the quadratic expression:
(x + 4)(x - 1) = 0
Using the zero product property, we have two possible solutions for x:
x + 4 = 0 or x - 1 = 0
Solving for x in each equation gives us:
x = -4 or x = 1
Since the length and width cannot be negative, we discard the solution x = -4.
Therefore, the length of the rectangular prism is x + 2, which is equal to 1 + 2 = 3, and the width is x + 1, which is equal to 1 + 1 = 2.
Hence, the length of the rectangular prism is 3 and the width is 2.