Problem 1.) Three friends want to share a circular 16-inch pizza equally by exactly only two parallel cuts. How far from the center must the cuts be? Hint: The pizza is a circle of radius 8 inches.

Problem 2.) You are at the southernmost point of a circular lake of radius 8 miles. Your plan is to swim a straight course to another point on the shore of the lake, then jog to the northernmost point. You can jog 150% as fast as you can swim. With a detailed explanation of your steps, find, in minutes, the least and the most time this trip will take if you swim at 3.5 miles per hour.

since the area of a circular segment is

a = 1/2 r^2 (θ-sinθ)

you need θ such that cos(θ/2) = x/8
so, you need to solve

1/2 * 64 (2*arccos(x/8)-sin(2*arccos(x/8))) = 64π/3
x = 2.12 inches
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3.5 mi/hr = 0.058 mi/min
If the swim subtends an angle θ, then the distance swum is 2x, then

x/8 = sin θ/2
x = 8 sinθ/2
the distance jogged is then 8π - 8θ

total time is then

(16 sin(θ/2))/.0583 + (8π - 8θ)/.0875

If you poke around at the graph, you can prolly figure the max/min for the domain 0 <= θ <= π

http://www.wolframalpha.com/input/?i=%2816+sin%28%CE%B8%2F2%29%29%2F.0583+%2B+%288%CF%80+-+8%CE%B8%29%2F.0875

LOL ITS 33,3

Problem 1:

To find the distance from the center where the two cuts should be made, we need to divide the pizza into three equal parts. Since the pizza is a circle with a radius of 8 inches, the circumference of the whole pizza is given by the formula 2 * π * r, where r is the radius.

Circumference = 2 * π * 8 inches = 16π inches (approximately 50.27 inches)

To divide the circumference equally between the three friends, we need to find the length of each section. This can be done by dividing the total circumference by 3:

Length of each section = Circumference / 3 = (16π inches) / 3 (approximately 16.76 inches)

Therefore, the distance from the center where the cuts should be made is half the length of each section. Hence,

Distance from center = (Length of each section) / 2 = (16.76 inches) / 2 (approximately 8.38 inches)

So, the cuts should be made approximately 8.38 inches from the center.

Problem 2:

To find the least and the most time for the trip, we need to consider the time it takes to swim the distance and the time it takes to jog the remaining distance.

Let's calculate the least and the most distances to be covered.

Least distance: Since the starting point is the southernmost point, the least distance is the diameter of the lake, which is twice the radius.

Least distance = 2 * radius = 2 * 8 miles = 16 miles

Most distance: The longest distance is when you swim in a straight line to a point on the shoreline directly north of the starting point, and then jog to the northernmost point on the shoreline. This can be found using the Pythagorean theorem.

Let x be the distance you swim in miles before reaching the shoreline. Then, the most distance can be calculated as follows:

(x^2) + (8 miles)^2 = (16 miles)^2

Simplifying the equation:

x^2 + 64 = 256

x^2 = 256 - 64

x^2 = 192

x ≈ √192 ≈ 13.86 (approximately)

The most distance is approximately 13.86 miles.

Now, let's calculate the time it takes to swim and jog for both the least and the most distances.

Time to swim = Distance / Speed

Least time to swim = 16 miles / 3.5 miles per hour = 4.57 hours

Most time to swim = 13.86 miles / 3.5 miles per hour ≈ 3.96 hours

Time to jog = Distance / Speed

Least time to jog = (16 miles - 16 miles) / (1.5 * 3.5 miles per hour) = 0 hours

Most time to jog = (13.86 miles - 16 miles) / (1.5 * 3.5 miles per hour) ≈ -0.21 hours

Since it is not possible to have negative time, the most time to jog is 0 hours.

Therefore, the least time for the trip is approximately 4.57 hours or 4 hours and 34 minutes, and the most time for the trip is approximately 3.96 hours or 3 hours and 57 minutes.