Simon and his track team ran on the Heron Peak running trail yesterday after school. The team ran 2 miles due west from the parking lot to a bench. They turned at the bench to run 1.5 miles due north toward an outhouse. To finish the run, the team ran a straight line from the outhouse back to the parking lot. If Simon ran at a constant rate of 8 miles per hour, how long did it take Simon to finish the run?
If necessary, round your answer to the nearest tenth.
To find out how long it took Simon to finish the run, we can first figure out the total distance of the run and then calculate the time it took him running at a speed of 8 miles per hour.
Simon and his track team ran the following segments:
1. 2 miles due west
2. 1.5 miles due north
3. A straight line back to the parking lot
The first two segments form two sides of a right-angled triangle, with the straight line back to the parking lot being the hypotenuse. We can use the Pythagorean theorem to find the length of this hypotenuse, which is the distance from the outhouse back to the parking lot.
The Pythagorean theorem states that in a right-angled triangle:
c² = a² + b²
where c is the length of the hypotenuse, a is one side, and b is the other side of the triangle. Here, a is 2 miles, and b is 1.5 miles. Plugging these values into the theorem gives us:
c² = (2 miles)² + (1.5 miles)²
c² = 4 + 2.25
c² = 6.25
So, the hypotenuse (c) is:
c = √6.25
c = 2.5 miles
The total distance of the run is the sum of all three segments:
Total distance = 2 miles + 1.5 miles + 2.5 miles
Total distance = 6 miles
Now we need to find out how long it took Simon to run 6 miles at a speed of 8 miles per hour.
Time = Distance / Speed
Time = 6 miles / 8 miles per hour
Time = 0.75 hours
Since we want the time in minutes, we can convert hours to minutes (there are 60 minutes in an hour):
Time in minutes = 0.75 hours × 60 minutes/hour = 45 minutes
Therefore, it took Simon 45 minutes to finish the run.
s=(2,6)
t=(5,2)
u=(-7,-7)
Right triangle △STU is shown on the coordinate plane below. ∠T is the right angle.
What is the area of △STU? If necessary, round your answer to the nearest tenth.
Given the coordinates of the vertices of right triangle △STU are S(2,6), T(5,2), and U(-7,-7), and knowing that ∠T is the right angle, we can find the lengths of the sides ST and TU to calculate the area of the triangle.
First, we'll find the length of ST using the distance formula between two points in the coordinate plane, which is:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
For ST:
\( x_1 = 2, y_1 = 6, x_2 = 5, y_2 = 2 \)
\( ST = \sqrt{(5 - 2)^2 + (2 - 6)^2} \)
\( ST = \sqrt{(3)^2 + (-4)^2} \)
\( ST = \sqrt{9 + 16} \)
\( ST = \sqrt{25} \)
\( ST = 5 \)
Now, we'll find the length of TU:
\( x_1 = 5, y_1 = 2, x_2 = -7, y_2 = -7 \)
\( TU = \sqrt{(-7 - 5)^2 + (-7 - 2)^2} \)
\( TU = \sqrt{(-12)^2 + (-9)^2} \)
\( TU = \sqrt{144 + 81} \)
\( TU = \sqrt{225} \)
\( TU = 15 \)
The area (A) of the right triangle is given by:
\[ A = \frac{1}{2} \times \text{base} \times \text{height} \]
In this scenario, ST and TU act as the base and height of the right triangle. So:
\[ A = \frac{1}{2} \times ST \times TU \]
\[ A = \frac{1}{2} \times 5 \times 15 \]
\[ A = \frac{1}{2} \times 75 \]
\[ A = 37.5 \]
Therefore, the area of triangle △STU is 37.5 square units. There's no need for rounding since the result is already to the nearest tenth.
Find the distance between the points (
–
7,
–
6) and (1,9). If necessary, round your answer to the nearest tenth.
To find the distance between the points (-7, -6) and (1, 9), you can use the distance formula, which is derived from the Pythagorean theorem and given by:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
Where \( (x_1, y_1) \) and \( (x_2, y_2) \) are the coordinates of the two points.
Let \( (x_1, y_1) = (-7, -6) \) and \( (x_2, y_2) = (1, 9) \).
Now plug in the values of the coordinates into the distance formula:
\[ d = \sqrt{(1 - (-7))^2 + (9 - (-6))^2} \]
\[ d = \sqrt{(1 + 7)^2 + (9 + 6)^2} \]
\[ d = \sqrt{(8)^2 + (15)^2} \]
\[ d = \sqrt{64 + 225} \]
\[ d = \sqrt{289} \]
\[ d = 17 \]
The distance between the points (-7, -6) and (1, 9) is 17 units. There's no need for rounding since the result is an exact whole number.
To find the time it took Simon to finish the run, we need to calculate the total distance he ran and then divide it by his constant rate of 8 miles per hour.
First, let's calculate the distance Simon ran from the parking lot to the bench. He ran 2 miles due west.
Next, let's calculate the distance Simon ran from the bench to the outhouse. He ran 1.5 miles due north.
To find the distance Simon ran from the outhouse back to the parking lot, we need to find the length of the straight line connecting the two points. This can be done using the Pythagorean theorem since we have a right triangle formed by the two legs (the distances Simon ran from the bench to the parking lot and the outhouse to the parking lot) and the hypotenuse (the straight line connecting the two points).
Using the Pythagorean theorem, we can find the length of the straight line:
c^2 = a^2 + b^2
where c is the length of the straight line, a is the distance Simon ran from the bench to the parking lot (2 miles), and b is the distance Simon ran from the outhouse to the parking lot (1.5 miles).
c^2 = 2^2 + 1.5^2
c^2 = 4 + 2.25
c^2 = 6.25
c ≈ 2.5 miles (rounded to the nearest tenth)
Now, let's calculate the total distance Simon ran by adding up the three distances:
2 miles + 1.5 miles + 2.5 miles = 6 miles
Finally, we can calculate the time it took Simon to finish the run by dividing the total distance (6 miles) by his constant rate (8 miles per hour):
Time = Distance / Rate
Time = 6 miles / 8 miles per hour
Time = 0.75 hours
Therefore, it took Simon approximately 0.75 hours (or 45 minutes) to finish the run.