A circular wheel of area \(1.54\,\text{m}^2\) rolls a distance of \(176\,\text{m}\). Find the number of revolutions made by the wheel.
40 revolutions
Step 1: Recall the formula for the area of a circle.
The area of a circle is given by \(A = \pi r^2\), where \(r\) is the radius.
Step 2: Substitute the given area.
We are told that the area of the wheel is \(1.54\,\text{m}^2\).
So, \(\pi r^2 = 1.54\).
Step 3: Solve for \(r^2\).
\(r^2 = \dfrac{1.54}{\pi}\).
Taking \(\pi \approx 3.14\):
\(r^2 = \dfrac{1.54}{3.14} \approx 0.49\).
Step 4: Find the radius.
\(r = \sqrt{0.49} = 0.7\,\text{m}\).
Step 5: Find the circumference of the wheel.
The circumference (distance covered in one revolution) is \(C = 2\pi r\).
\(C = 2 \times 3.14 \times 0.7 \approx 4.4\,\text{m}\).
Step 6: Calculate the number of revolutions.
Total distance travelled = \(176\,\text{m}\).
Number of revolutions = \(\dfrac{\text{Total distance}}{\text{Circumference}}\).
\(= \dfrac{176}{4.4} = 40\).
Final Answer: The wheel makes 40 revolutions.