Question

Find the arc length from (-3,4) clockwise to (4,3) along the circle \(x^{2}+y^{2}=25\). Show that the result is one-fourth the circumference of the circle.

Short answer

The arc length between the two points is approximately 8.8685 units which does not equal to a quarter of the circumference which is approximately 7.8535 units.

Step-by-step solution

Calculate the radius of the circle

The equation of the circle is \(x^{2}+y^{2}=25\), which shows that the radius \(r=\sqrt{25}=5\).

Find the angles corresponding to points

The two points are (-3,4) and (4,3). Converting these to their corresponding polar angles we get \(\tan^{-1}(-4/3)\) and \(\tan^{-1}(3/4)\). The inverse tangent gives angles in the range \(-\pi/2\) to \(+\pi/2\), take these as initial angles which come out to be approximately -0.93 radians and 0.6435 radians respectively. Thus we need to adjust these angles to their appropriate quadrants. With some thought, it can be seen that the angle for (-3,4) should be \(\pi - 0.93\), and the one for (4,3) should be \(0.6435\), in radians.

Computing the arc length

The arc length is given by the difference between the angles multiplied by radius. Therefore, the arc length from (-3,4) to (4,3) is measured clockwise along the circle, which is \((0.6435 - (\pi - 0.93)) \times 5 \) radians, and approximately 8.8685 units.

Show that the arc length is one-fourth the circumference

The circumference of the circle with radius 5 is \( 2\pi \times 5 = 10\pi \). Dividing this by 4, we obtain \( \frac{10\pi}{4} \). This is approximately 7.8535. Thus the arc length calculated in Step 3 is more than one-fourth the circumference. The final step does not match with the originally stated problem. This might indicate an error in the problem or in the arc length calculation.