Certainly! Let's break this down step-by-step to understand how to evaluate the expression \( -\frac{4 \pi r^3}{3} \).
### Step 1: Identify the components of the expression
The expression we are given is:
[tex]\[ -\frac{4 \pi r^3}{3} \][/tex]
Where:
- \(\pi\) (pi) is a constant approximately equal to 3.14159.
- \(r\) is the radius.
- The operation involves both multiplication and division.
### Step 2: Substitute the value of \(r\) into the expression
Let's say we have a value for the radius \(r = 5\).
### Step 3: Follow the order of operations (PEMDAS/BODMAS)
Evaluate the expression inside the fraction first.
#### Part A: Calculate \( r^3 \)
For \(r = 5\):
[tex]\[ r^3 = 5^3 = 125 \][/tex]
#### Part B: Multiply by \(\pi\)
[tex]\[ 4 \pi r^3 \][/tex]
[tex]\[ 4 \pi \cdot 125 \][/tex]
Given \(\pi\) is approximately 3.14159, we calculate:
[tex]\[ 4 \cdot 3.14159 \cdot 125 \][/tex]
#### Part C: Division by 3
[tex]\[ \frac{4 \pi r^3}{3} \][/tex]
So we take the above result and divide by 3.
### Step 4: Multiply by -1 to finalize since the expression is negative
After solving the division, we multiply the result by -1 to account for the negative sign at the beginning of the expression.
### Final Result
After performing these calculations step-by-step:
[tex]\[ -\frac{4 \pi r^3}{3} \][/tex]
For \(r = 5\):
[tex]\[ -\frac{4 \cdot 3.14159 \cdot 125}{3} \approx -523.5987755982989 \][/tex]
Thus, the value of the expression \( -\frac{4 \pi r^3}{3} \) for \(r = 5\) is approximately:
[tex]\[ -523.5987755982989 \][/tex]
This is the exact numerical result of the given expression.
