Question

jennifer has estimated the volume of this cylinder to be \\(1500\text{ cm}^3\\).

a) show the workings used to calculate this estimate.

b) will the real volume be larger or smaller than \\(1500\text{ cm}^3\\)? write an explanation.

Explanation:

⚡ Using what you learned: volume of prisms, pyramids, cylinders, cones, spheres · Estimation and Rounding in Computation

Step 1: Identify the dimensions and formula

The cylinder has:

• Diameter \( d = 10\text{ cm} \), which means the radius is:

$$ r = \frac{10}{2} = 5\text{ cm} $$

• Height \( h = 20\text{ cm} \)

The formula for the volume of a cylinder is:

$$ V = \pi r^2 h $$

Step 2: Show the workings for the estimate

To estimate the volume without a calculator, we round the value of \(\pi \approx 3.14159...\) to \( 3 \):

$$ V \approx 3 \times 5^2 \times 20 $$

$$ V \approx 3 \times 25 \times 20 $$

$$ V \approx 3 \times 500 = 1500\text{ cm}^3 $$

Step 3: Determine if the real volume is larger or smaller

The real value of \(\pi\) is approximately \( 3.14 \), which is larger than the estimated value of \( 3 \).

Since we underestimated the value of \(\pi\) in our calculation:

$$ 3.14 > 3 $$

The real volume will be larger than the estimated volume of \( 1500\text{ cm}^3 \).

Answer:

a)

$$ \text{Radius } (r) = 5\text{ cm} $$

$$ \text{Using } \pi \approx 3: $$

$$ \text{Volume} \approx 3 \times 5^2 \times 20 = 3 \times 25 \times 20 = 1500\text{ cm}^3 $$

b)
The real volume will be larger because \(\pi\) was rounded down to \(3\) for the estimate. Since the actual value of \(\pi\) (\(\approx 3.14\)) is larger than \(3\), the actual volume will be larger than \(1500\text{ cm}^3\).