Question

find the surface area and volume of sphere. round answer to 2 decimal places if needed. surface area of sphere = 380.13 × volume of sphere = 706.86 × question help: read video message instructor submit question

Explanation:

Step1: Identify the radius

The diameter of the sphere is given as 11, so the radius \( r=\frac{11}{2} = 5.5\).

Step2: Calculate Surface Area of Sphere

The formula for the surface area of a sphere is \( SA = 4\pi r^{2}\).
Substitute \( r = 5.5\) into the formula:
\( SA=4\pi(5.5)^{2}=4\pi\times30.25 = 121\pi\approx121\times3.1416\approx380.13\) (Wait, but let's recalculate properly. Wait, \( 5.5^2 = 30.25\), \( 4\times30.25=121\), \( 121\times\pi\approx121\times3.14159265\approx380.1327\), which is approximately 380.13. But maybe the user made a mistake in the previous answer? Wait, no, maybe the diameter is 11, so radius is 5.5. Wait, maybe the original problem had a different diameter? Wait, no, the diagram shows the distance from the center to the edge (radius) or the diameter? Wait, the red line is from the edge to the center, so that's the radius? Wait, no, the red line is from one edge through the center to the other edge? Wait, the dot is the center, so the red line from the edge to the center is the radius. Wait, the length is 11? Wait, that would mean the radius is 11? Wait, maybe I misread. Let me check again. The diagram: the red line is from the left edge to the center, labeled 11. So that's the radius \( r = 11\)? Oh! I made a mistake earlier. If the red line is from the edge to the center, that's the radius, so \( r = 11\). Oh! That's the error. So let's correct that.

Step1 (Corrected): Identify the radius

The radius \( r = 11\) (since the red line is from the edge to the center, so that's the radius).

Step2 (Corrected): Calculate Surface Area of Sphere

The formula for the surface area of a sphere is \( SA = 4\pi r^{2}\).
Substitute \( r = 11\) into the formula:
\( SA = 4\pi(11)^{2}=4\pi\times121 = 484\pi\approx484\times3.1416\approx1520.53\)

Step3: Calculate Volume of Sphere

The formula for the volume of a sphere is \( V=\frac{4}{3}\pi r^{3}\).
Substitute \( r = 11\) into the formula:
\( V=\frac{4}{3}\pi(11)^{3}=\frac{4}{3}\pi\times1331=\frac{5324}{3}\pi\approx\frac{5324}{3}\times3.1416\approx\frac{5324\times3.1416}{3}\approx\frac{16723.38}{3}\approx5574.46\)

Wait, but the original answers were wrong because the radius was misidentified. Let's confirm: if the red line is from the edge to the center, that's the radius. So if the length is 11, then radius \( r = 11\). Let's recalculate:

Surface Area: \( 4\pi r^2 = 4\times\pi\times11^2 = 4\times\pi\times121 = 484\pi \approx 484\times3.1415926535 \approx 1520.53\)

Volume: \( \frac{4}{3}\pi r^3 = \frac{4}{3}\times\pi\times11^3 = \frac{4}{3}\times\pi\times1331 = \frac{5324}{3}\pi \approx 5575.28\) (rounded to two decimal places)

Ah, so the initial mistake was taking the radius as 5.5 instead of 11. The red line is from the edge to the center, so that's the radius, length 11.

So let's redo the steps correctly:

Step1: Determine the radius

From the diagram, the radius \( r = 11\) (the distance from the center to the surface of the sphere).

Step2: Calculate Surface Area

Use the formula \( SA = 4\pi r^2 \).
Substitute \( r = 11 \):
\( SA = 4 \times \pi \times 11^2 = 4 \times \pi \times 121 = 484\pi \approx 484 \times 3.14159265 \approx 1520.53 \)

Step3: Calculate Volume

Use the formula \( V = \frac{4}{3}\pi r^3 \).
Substitute \( r = 11 \):
\( V = \frac{4}{3} \times \pi \times 11^3 = \frac{4}{3} \times \pi \times 1331 = \frac{5324}{3}\pi \approx \frac{5324 \times 3.14159265}{3} \approx \frac{16723.38}{3} \approx 5574.46 \) (or more accurately, \( \frac{4}{3}\times1331\times\pi = \frac{5324}{3}\pi \approx 5575.28 \) when calculated with more precise \(\pi\))

Wait, let's use a calculator for volume:

\( 11^3 = 1331 \)

\( \frac{4}{3} \times 1331 = \frac{5324}{3} \approx 1774.6667 \)

\( 1774.6667 \times \pi \approx 1774.6667 \times 3.14159265 \approx 5575.28 \)

So the correct surface area is approximately 1520.53 and volume is approximately 5575.28.

Answer:

Surface Area of Sphere = \(\boxed{1520.53}\)
Volume of Sphere = \(\boxed{5575.28}\)