Question

1. the circle has a radius of 3 units. three students tried to calculate the area. order their answers from the farthest from the exact to closest to the exact area. 9π units 28.26π units 28.26 units

Explanation:

Step1: Recall the area formula of a circle

The formula for the area of a circle is \( A=\pi r^{2} \). Given the radius \( r = 3 \) units, the exact area is \( A=\pi\times3^{2}=9\pi \) square units.

Step2: Analyze each student's answer

• First answer: \( 9\pi \) units. This is the exact area, so it is the closest to the exact value.
• Second answer: \( 28.26\pi \) units. Let's find the difference from the exact area. The exact area is \( 9\pi \), so the difference is \( 28.26\pi- 9\pi=19.26\pi \), which is a large difference.
• Third answer: \( 28.26 \) units. We know that \( \pi\approx3.14 \), so the exact area \( 9\pi\approx9\times3.14 = 28.26 \) units. So this answer is an approximation of the exact area (using \( \pi\approx3.14 \)), so its difference from the exact area is very small (close to 0). Wait, no, wait: Wait, the exact area is \( 9\pi \approx28.26 \) (when \( \pi = 3.14 \)). Wait, let's re - evaluate:

Wait, the exact area is \( A = \pi r^{2}=9\pi\approx28.26 \) (if \( \pi = 3.14 \)). Let's list the three answers:

1. \( 9\pi \): Exact value.
2. \( 28.26\pi \): Let's compute its numerical value. If \( \pi\approx3.14 \), then \( 28.26\pi\approx28.26\times3.14 = 88.7364 \)
3. \( 28.26 \): Which is approximately \( 9\pi \) (since \( 9\times3.14 = 28.26 \))

So the exact area is \( 9\pi\approx28.26 \). Now let's find the absolute differences from the exact area (\( 9\pi\approx28.26 \)):

• For \( 28.26\pi \): \( |28.26\pi - 9\pi|=19.26\pi\approx19.26\times3.14 = 60.4764 \)
• For \( 28.26 \): \( |28.26 - 9\pi|=|28.26 - 28.26| = 0 \) (when \( \pi = 3.14 \))
• For \( 9\pi \): \( |9\pi - 9\pi| = 0 \) (exact)

Wait, there is a mistake in the initial analysis. Let's correct it:

The exact area of the circle with radius \( r = 3 \) is \( A=\pi r^{2}=9\pi\approx28.26 \) (when \( \pi = 3.14 \)).

Now let's analyze each answer:

• Answer 1: \( 9\pi \): This is the exact area.
• Answer 2: \( 28.26\pi \): This is a much larger value. Let's calculate the difference from the exact area: \( 28.26\pi-9\pi = 19.26\pi\approx19.26\times3.14 = 60.4764 \)
• Answer 3: \( 28.26 \): Since \( 9\pi\approx28.26 \) (when \( \pi = 3.14 \)), this is an approximation of the exact area (using \( \pi = 3.14 \)). So the difference between \( 28.26 \) and \( 9\pi \) is \( |28.26 - 9\pi|=|28.26 - 28.26| = 0 \) (when \( \pi = 3.14 \)). Wait, but \( 9\pi \) is the exact value, and \( 28.26 \) is an approximate value of \( 9\pi \) (using \( \pi\approx3.14 \)).

So to order from farthest to closest:

The farthest from the exact area is \( 28.26\pi \) (because its difference from \( 9\pi \) is \( 19.26\pi \), which is very large), then we need to compare \( 28.26 \) and \( 9\pi \). Wait, \( 9\pi \) is the exact value, and \( 28.26 \) is an approximation of \( 9\pi \) (with \( \pi\approx3.14 \)). So the distance of \( 28.26 \) from \( 9\pi \) is \( |28.26 - 9\pi|=|9\times3.14 - 9\pi| = 9|3.14-\pi| \), and the distance of \( 9\pi \) from itself is 0. So actually, \( 9\pi \) is the exact value (distance 0), and \( 28.26 \) has a small distance (since \( 3.14 \) is a common approximation of \( \pi \)).

Wait, the problem says "order their answers from the farthest from the exact to closest to the exact area".

Exact area: \( A = 9\pi \)

Answer:

1: \( 9\pi \): distance 0