Length = 4 + x Width = x Height = x2 + 1 What is the base area of Box 3?
The correct answer and explanation is:
To find the base area of Box 3, we are given the following:
- Length = 4+x4 + x
- Width = xx
- Height = x2+1x^2 + 1
However, the question only asks for the base area, which is the area of the rectangle formed by the length and width.
Step 1: Formula for Base Area
The base area (A) of a rectangular box is calculated by: A=Length×WidthA = \text{Length} \times \text{Width}
Step 2: Plug in the Given Expressions
Using the expressions provided: A=(4+x)×xA = (4 + x) \times x
Step 3: Simplify the Expression
Now multiply the binomial: A=x(4+x)=4x+x2A = x(4 + x) = 4x + x^2
So, the base area of Box 3 is: x2+4x\boxed{x^2 + 4x}
300-Word Explanation:
To determine the base area of a rectangular box, you only need the length and width measurements, because the base is simply a flat rectangle. In this problem, we are given expressions rather than specific numbers:
- The length of the box is 4+x4 + x,
- The width is xx,
- And while the height is given as x2+1x^2 + 1, it is not relevant for finding the base area.
The base area of any rectangle is calculated by multiplying the length by the width. That’s why we take (4+x)×x(4 + x) \times x. This is an example of using the distributive property in algebra: multiplying a monomial by a binomial.
So, (4+x)×x=4x+x2(4 + x) \times x = 4x + x^2
This result, x2+4xx^2 + 4x, is an algebraic expression that represents the area of the base in terms of the variable xx. It means that for any value of xx, you can plug it into this expression to find the specific area. For example, if x=2x = 2, then the base area would be: 22+4(2)=4+8=122^2 + 4(2) = 4 + 8 = 12
In conclusion, the base area is not a fixed number but an expression that depends on xx. The correct answer is: x2+4x\boxed{x^2 + 4x}