From a window in Academy 7, one can observe the front door of Phillips Hall, which is about 45 yards away. A turn of the head through an angle of 63 degrees in a counterclockwise direction enables one to see the front door to Jeremiah Smith Hall which is about 65 yards away. If you spot a friend coming out of that door, and if that friend proceeds directly to the Phillips Hall door, making the trip in 48 seconds, what would you estimate to be your friend's average speed during the walk?
To estimate the friend's average speed during the walk, we need to calculate the distance they walked and divide it by the time it took.
First, we can use trigonometry to find the distance the friend walked to reach the Phillips Hall door. Since the friend moved in a counterclockwise direction, we know that the angle formed by the line connecting the two doors and the line connecting the window and the Phillips Hall door is 63 degrees. Using the sine function, we can find the distance as follows:
Distance to Phillips Hall = 45 yards / sin(63 degrees) = 52.74 yards (approximately)
Next, we divide this distance by the time it took the friend to walk to find the average speed:
Average speed = Distance / Time = 52.74 yards / 48 seconds = 1.09875 yards per second (approximately)
Therefore, we would estimate the friend's average speed during the walk to be approximately 1.09875 yards per second.
use the cosine rule, the triangle is not a right triangle
Apologies for the confusion. You are correct that we cannot directly use the sine function since the triangle is not a right triangle.
To find the distance the friend walked, we can use the cosine rule. Let's define the angle formed by the line connecting the two doors, the line connecting the window and the Phillips Hall door as A, the distance from the window to the Phillips Hall door as x, and the distance from the window to the Jeremiah Smith Hall door as y.
According to the cosine rule, we have:
x^2 = 45^2 + 65^2 - 2 * 45 * 65 * cos(A)
Now, we know that the friend made the trip in 48 seconds, so their average speed can be calculated by dividing the total distance by the time:
Average speed = (x + y) / 48 seconds
To find y, we can use the sine rule. The angle formed by the line connecting the two doors, the line connecting the window and the Jeremiah Smith Hall door, and the angle A is supplementary to each other. So we have:
sin(A) = y / 65
Rearranging this equation, we get:
y = sin(A) * 65
Substituting this into the average speed equation, we have:
Average speed = (x + sin(A) * 65) / 48 seconds
To solve these equations, we need the value of angle A. Unfortunately, the problem statement does not provide this information, so we cannot make an accurate estimation of the friend's average speed without further data.
A = 63
Apologies for missing that information. With A = 63 degrees, we can now proceed with the calculations.
Using the cosine rule, we have:
x^2 = 45^2 + 65^2 - 2 * 45 * 65 * cos(63 degrees)
x ≈ 78.95 yards
Now, let's find y using the sine rule:
sin(63 degrees) = y / 65
y ≈ sin(63 degrees) * 65 ≈ 57.27 yards
Average speed = (x + y) / 48 seconds
Average speed ≈ (78.95 + 57.27) / 48
Average speed ≈ 2.365 yards per second
Therefore, we would estimate your friend's average speed during the walk to be approximately 2.365 yards per second.
To estimate your friend's average speed during the walk from Jeremiah Smith Hall to the Phillips Hall door, we will need to calculate the distance and time taken for the walk. Here are the steps to do so:
Step 1: Find the distance between Jeremiah Smith Hall and the Phillips Hall door:
Using the given information, the distance from the window in Academy 7 to the Phillips Hall door is 45 yards, and the distance from the window in Academy 7 to the Jeremiah Smith Hall door is 65 yards.
Step 2: Calculate the walking distance:
To find the walking distance between the two doors, we need to use trigonometry. Given the angle of 63 degrees, we can calculate the horizontal distance using the trigonometric function cosine. Here's the formula to find the walking distance:
Walking distance = Distance from Academy 7 to Jeremiah Smith Hall door * cos(angle of 63 degrees)
Walking distance = 65 yards * cos(63 degrees)
Calculating this value, we get:
Walking distance = 65 yards * 0.437054
Walking distance ≈ 28.35 yards (rounded to two decimal places)
Step 3: Calculate the average speed:
The average speed is given by the formula:
Average Speed = Distance / Time
Given that the walking distance is approximately 28.35 yards and the time taken to walk is 48 seconds, we can calculate the average speed as follows:
Average Speed = 28.35 yards / 48 seconds
Calculating the average speed, we get:
Average Speed ≈ 0.5906 yards per second (rounded to four decimal places)
Therefore, your friend's estimated average speed during the walk from Jeremiah Smith Hall to the Phillips Hall door would be approximately 0.5906 yards per second.
To estimate your friend's average speed during the walk, we need to use the information provided and apply basic trigonometry.
First, let's understand the distances and angles involved:
- The distance between Academy 7 and Phillips Hall is 45 yards.
- The distance between Academy 7 and Jeremiah Smith Hall is 65 yards.
- The angle turned from Phillips Hall to Jeremiah Smith Hall is 63 degrees.
Now, let's break down the friend's walk into two parts:
1. Walking from Jeremiah Smith Hall to the position where you spotted them (at Academy 7).
2. Walking from the position where you spotted them to Phillips Hall.
To estimate the average speed, we need to calculate the time taken for each part.
Part 1: Walking from Jeremiah Smith Hall to the observed position:
Since the distance is 65 yards and the angle turned is 63 degrees, we can use trigonometry to calculate the horizontal distance covered (adjacent side) during this walk. The formula is:
Adjacent side = Hypotenuse * cos(angle)
Adjacent side = 65 yards * cos(63 degrees)
This gives us the horizontal distance covered, which is the same as the distance between Academy 7 and Jeremiah Smith Hall.
Part 2: Walking from the observed position to Phillips Hall:
The distance is given as 45 yards, and we know the time taken for this part of the walk is 48 seconds.
Now, to estimate the friend's average speed, we can add the distances from both parts and divide by the total time taken:
Average speed = Total distance / Total time
Total distance = Distance from Jeremiah Smith Hall to observed position + Distance from observed position to Phillips Hall
Total time = Time taken for the whole trip (48 seconds)
Plug in the values into the formula:
Average speed = (Distance J.Smith Hall to observed position + Distance observed position to Phillips Hall) / Total time
You can now calculate the average speed by substituting the values we obtained earlier and solving the equation to get the estimate of your friend's average speed during the walk.