I will be greatfull if you help me to solve the below problem:


The minute hand of a clock is 12cm long.
Find the area of the surface of the clock described by the minute hand in 35min.

270

total area = π(12^2) = 144π

area covered in 35 minutes = (35/60)(144π)
= 84π
= appr. 264 cm^2

To find the area of the surface described by the minute hand of a clock in 35 minutes, we need to divide the space covered by the minute hand into two parts: a sector of a circle and a triangle.

First, let's calculate the area of the sector of the circle. The minute hand moves in a circular path, and the length of the minute hand is equal to the radius of this circle, which is 12 cm.

The formula to calculate the area of a sector is:

Area of sector = (θ/360) x π x r^2

where θ is the central angle of the sector, π is a constant approximately equal to 3.14159, and r is the radius.

Since the minute hand has covered 35 minutes, we need to determine the corresponding central angle (θ). To find the angle, we can use the fact that a complete revolution of the minute hand takes 60 minutes, which corresponds to a 360° angle.

So, we can calculate the central angle (θ):

θ = (35/60) x 360

Next, we can substitute the calculated values of θ and r into the formula to find the area of the sector:

Area of sector = (θ/360) x π x r^2

Area of sector = ((35/60) x 360 / 360) x 3.14159 x 12^2

Now, let's move on to finding the area of the triangle formed by the minute hand. The base of the triangle is equal to the length of the minute hand, which is 12 cm. The height of the triangle can be calculated using basic trigonometry.

Since the angle at the center of the clock is the same as the central angle of the sector, we can use the same value of θ we calculated earlier.

The formula to calculate the area of a triangle is:

Area of triangle = (base x height)/2

So, substituting the values:

Area of triangle = (12 x sin(θ)) / 2

Now, add the areas of the sector and the triangle to find the total area covered by the minute hand:

Total area covered = Area of sector + Area of triangle

Finally, calculate the values using a calculator or by substituting the calculated value of θ into the formulas and simplify to find the solution.