# a student strolls diagnally across a level rectangular campus plaza, covering the 50 m distance in 1.0 min (a) if the diagonal route makes a 37 degree angle with the long side of the plaza, what would be the distance if the student had walked halfway around the outside of the plaza instead of along the diagonial route ? (b)if the student had walked the outside route in 1.0 min at a constant speed of how much time would she have spent on each side?

## (a) The short side of the rectangle will be b = 50 sin 37 meters. The long side is 50 cot 37. If a is the long side and b is the short side length, the distance walked around the edge, instead of the diagonal, is a + b.

(b) time = (side length)/speed
The speed is 50 m/mimute

## you are correct on A but B is wrong.

You need to convert the results for A and B. given that it is 70m/m forget about the initial distance 50m/m this is a whole separate question.

Now,

A (times) m/70m = .57m
It is a simple conversion from there.

## To solve this problem, we can use trigonometry and geometry concepts.

Let's start with part (a) and find the distance if the student had walked halfway around the outside of the plaza instead of along the diagonal route.

Step 1: Determine the length of the long side of the plaza.
Using the given information, we know that the student covers a distance of 50 m in 1.0 min. We have to find the length of the long side of the plaza.

We can use trigonometry to find the length. Given that the diagonal makes a 37 degree angle with the long side, we can use the cosine function:

cos(37 degrees) = adjacent / hypotenuse

cos(37 degrees) = long side / 50 m

Simplifying:

long side = 50 m * cos(37 degrees)

Step 2: Calculate the distance if the student walked halfway around the outside of the plaza.
Since the student walks halfway around the outside of the plaza, they would cover three sides of the plaza: the long side, the short side, and the other long side.

Therefore, the distance would be:

distance = 2 * (long side) + (short side)

Step 3: Find the length of the short side.
Since the plaza is a rectangle, we know that the short side has the same length as the diagonal's perpendicular side.

sin(37 degrees) = opposite / hypotenuse

sin(37 degrees) = short side / 50 m

Simplifying:

short side = 50 m * sin(37 degrees)

Now we can calculate the total distance covered:

distance = 2 * (long side) + (short side)
distance = 2 * (50 m * cos(37 degrees)) + (50 m * sin(37 degrees))

For part (b), we need to find how much time the student would have spent on each side if she had walked the outside route in 1.0 min at a constant speed.

Since the student walks along each side with a constant speed, the time spent on each side would be proportional to the length of the side. So, we can find the ratio of the distance covered on each side to the total distance, and multiply that by the total time of 1.0 min to find the time spent on each side.

time on each side = (distance on each side / total distance) * 1.0 min

Please provide the values for the angle (b)student had walked the outside route in 1.0 min at a constant speed to proceed with the calculation.

## To solve this problem, we can make use of the Pythagorean theorem and trigonometric concepts. Let's first start with part (a) to determine the distance if the student had walked halfway around the outside of the plaza instead of along the diagonal route.

(a) To find the distance if the student had walked halfway around the outside of the plaza, we need to calculate the length of both sides of the rectangular plaza.

Let's denote the length of the longer side of the plaza as L and the shorter side as W. Since we are given the diagonal route and the angle it makes with the long side of the plaza, we can use trigonometry to find the length of the longer side.

Using the given angle, which is 37 degrees, we can use the sine function to find the length of the longer side (L):

sin(37 degrees) = L / 50m

Rearranging the equation, we get:

L = 50m * sin(37 degrees)

Now that we have the length of the longer side, we need to find the length of the shorter side, which can be done using the Pythagorean theorem.

Using the formula for the Pythagorean theorem, we have:

L^2 = (L/2)^2 + W^2

Simplifying the equation, we get:

3L^2 / 4 = W^2

Now, substituting the value of L from the earlier equation, we have:

3(50m * sin(37 degrees))^2 / 4 = W^2

Simplifying further:

W^2 = 2250m^2 * sin^2(37 degrees)

Finally, we can find the distance if the student had walked halfway around the outside of the plaza by adding the length of both sides:

Distance = 2W

(b) To determine the time spent on each side when walking the outside route at a constant speed, we should first calculate the total distance covered.

The total distance covered is the perimeter of the rectangular plaza, which is given by:

Perimeter = 2L + 2W

Since the student takes 1.0 min to walk the outside route at a constant speed, and we know that Speed = Distance / Time, we can rearrange the equation to find the time spent on each side:

Time spent on each side = (Distance of each side) / (Speed)

Now, substituting the value of Distance of each side as W, we have:

Time spent on each side = W / (Speed)

Remember, Speed = Perimeter / Time

Time spent on each side = W / (Perimeter / Time)

Simplifying further:

Time spent on each side = (Time * W) / Perimeter

Substituting the values we have:

Time spent on each side = (1.0 min * W) / (2L + 2W)

Now, using the earlier calculations for L and W, we can substitute their values and calculate the time spent on each side.

Note: Make sure to convert the angle to radians if your calculator is set to radians mode. The formula for converting from degrees to radians is: radians = degrees * (pi/180).