Let's carefully analyze Lara's work and correct any mistakes she might have made.
1. Lara used the formula for the volume of a sphere: [tex]\(V = \frac{4}{3} \pi r^3\)[/tex].
2. However, she used the diameter ([tex]\(d = 10\)[/tex]) directly in the formula instead of the radius ([tex]\(r\)[/tex]). The radius of a sphere is half of the diameter. So for a diameter of 10, the radius [tex]\(r = \frac{10}{2} = 5\)[/tex].
3. Rewriting the volume formula with the radius, we get:
[tex]\[ V = \frac{4}{3} \pi r^3 \][/tex]
Substituting [tex]\( r = 5 \)[/tex], we get:
[tex]\[ V = \frac{4}{3} \pi (5)^3 \][/tex]
[tex]\[ V = \frac{4}{3} \pi (125) \][/tex]
[tex]\[ V = \frac{500}{3} \pi \][/tex]
Let’s address the specific steps Lara took and identify her mistakes:
1. Lara’s incorrect step was:
[tex]\[ V = \frac{4}{3} \pi (10)^3 \][/tex]
Here, Lara incorrectly used the diameter instead of the radius. She should have used the radius, which is 5.
2. The next step in Lara’s incorrect calculation was:
[tex]\[ V = \frac{4}{3} \pi (1000) \][/tex]
Lara continued with the diameter, resulting in an incorrect volume calculation.
The correct calculation should follow these steps:
[tex]\[ V = \frac{4}{3} \pi (5)^3 \][/tex]
[tex]\[ V = \frac{4}{3} \pi (125) \][/tex]
[tex]\[ V = \frac{500}{3} \pi \][/tex]
Evaluating this, we find:
[tex]\[ V \approx 523.5987755982989 \, \text{units}^3 \][/tex]
Thus, Lara’s corrected volume should be approximately [tex]\(523.5987755982989 \, \text{units}^3\)[/tex].
#### Answer to Specific Statements:
1. [tex]\( V = \frac{4}{3} \pi (1000) \)[/tex]
- No, she used the diameter instead of the radius.
2. [tex]\( V = \frac{4000}{3} \pi \, \text{units}^3 \)[/tex]
- No, her mistake was in using the diameter instead of the radius.
- She should have used the correct radius and exponent, which would have led to the correct formula: [tex]\( V = \frac{500}{3} \pi \)[/tex].
