Let's work through the problem step by step to find an algebraic expression for the volume of the pyramid.
### Step 1: Identify the Variables
- Height (h) of the pyramid: [tex]\( h = x \)[/tex] units
- Length (l) of the base: [tex]\( l = x + 5 \)[/tex] units
- Width (w) of the base: [tex]\( w = x - \frac{1}{2} \)[/tex] units
### Step 2: Write the Formula for the Volume of a Pyramid
The volume [tex]\( V \)[/tex] of a right rectangular pyramid can be calculated by using the formula:
[tex]\[ V = \frac{1}{3} \times \text{base\_area} \times \text{height} \][/tex]
### Step 3: Calculate the Area of the Base
The area of the base of the pyramid, which is a rectangle, is given by:
[tex]\[ \text{base\_area} = l \times w \][/tex]
Substitute the values for length and width:
[tex]\[ \text{base\_area} = (x + 5) \left( x - \frac{1}{2} \right) \][/tex]
### Step 4: Simplify the Base Area Expression
Now, expand and simplify the multiplication:
[tex]\[ \text{base\_area} = (x + 5) \left( x - \frac{1}{2} \right) \][/tex]
[tex]\[ \text{base\_area} = x \left( x - \frac{1}{2} \right) + 5 \left( x - \frac{1}{2} \right) \][/tex]
[tex]\[ \text{base\_area} = x^2 - \frac{1}{2} x + 5x - \frac{5}{2} \][/tex]
[tex]\[ \text{base\_area} = x^2 + \frac{9}{2} x - \frac{5}{2} \][/tex]
### Step 5: Substitute the Simplified Base Area and Height into the Volume Formula
Now substitute the base area and the height into the volume formula:
[tex]\[ V = \frac{1}{3} \left( x^2 + \frac{9}{2} x - \frac{5}{2} \right) \times x \][/tex]
### Step 6: Simplify the Volume Expression
Multiply the expression and simplify:
[tex]\[ V = \frac{1}{3} \left( x^3 + \frac{9}{2} x^2 - \frac{5}{2} x \right) \][/tex]
[tex]\[ V = \frac{1}{3} x^3 + \frac{1}{3} \left( \frac{9}{2} \right) x^2 - \frac{1}{3} \left( \frac{5}{2} \right) x \][/tex]
[tex]\[ V = \frac{1}{3} x^3 + \frac{3}{2} x^2 - \frac{5}{6} x \][/tex]
### Step 7: Match the Expression with the Given Choices
Compare the obtained expression with the given choices:
[tex]\[ \frac{1}{3} x^3 + \frac{3}{2} x^2 - \frac{5}{6} x \][/tex]
This matches the first choice in the given options. So, the correct answer is:
[tex]\[ \boxed{\frac{1}{3} x^3 + \frac{3}{2} x^2 - \frac{5}{6} x} \][/tex]
