Jared takes a rectangular piece of fabric and makes a diagonal cut from one corner to the opposite corner. The cut he makes is 15 centimeters long and the width of the fabric is 9 centimeters. What is the fabric's length?
First, let's calculate the total distance traveled by the Island Loop Ferry along its route. The ferry travels in sections of straight lines that form a right triangle.
The first leg of the trip is due south from Seabreeze Village to Oceanview, which is 8 miles long. The second leg is due east from Oceanview to Portlandville, which is 15 miles long. These two legs form the perpendicular sides of a right triangle.
Now, to calculate the length of the third leg (the hypotenuse of the triangle), which is the direct trip back from Portlandville to Seabreeze Village, we use the Pythagorean theorem:
a^2 + b^2 = c^2
Where:
a = 8 miles (southward trip)
b = 15 miles (eastward trip)
c = the hypotenuse (the direct trip back)
Substitute the values and solve for c:
8^2 + 15^2 = c^2
64 + 225 = c^2
289 = c^2
Now take the square root of both sides:
c = √289
c = 17 miles
So, the ferry travels 8 miles south, then 15 miles east, and finally 17 miles diagonally back to Seabreeze Village. The total distance traveled for one complete route is the sum of all three legs:
Total distance = 8 + 15 + 17 = 40 miles
The ferry uses 1 gallon of fuel to travel 5 miles, so the fuel consumption for the entire route can be calculated by dividing the total distance by the distance traveled per gallon of fuel:
Fuel used = Total distance / Distance per gallon
Fuel used = 40 miles / 5 miles per gallon
Fuel used = 8 gallons
Therefore, the Island Loop Ferry uses 8 gallons of fuel to complete the route one time. There's no need to round since we have an exact answer.
When Jared makes a diagonal cut from one corner of the rectangular fabric to the opposite corner, he essentially creates two right-angled triangles. The diagonal cut represents the hypotenuse of these triangles, the width of 9 centimeters is one side (let's call it "a") and the length we're trying to determine is the other side (let's call it "b").
We can use the Pythagorean theorem to find the length of the rectangle since the triangle formed is a right triangle. The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse (c) is equal to the sum of the squares of the lengths of the other two sides (a and b).
So we have:
c^2 = a^2 + b^2
where c is the diagonal (15 cm), a is the width (9 cm), and b is the unknown length we want to find.
Now we plug in the values and solve for b:
15^2 = 9^2 + b^2
225 = 81 + b^2
b^2 = 225 - 81
b^2 = 144
Now we find the square root of both sides to solve for b:
b = √144
b = 12
Therefore, the length of the fabric is 12 centimeters.
Two cars leave the same parking lot, with one heading north and the other heading east. After several minutes, the eastbound car has traveled 3 miles. If the two cars are now a straight-line distance of 5 miles apart, how far has the northbound car traveled?
This problem can again be solved using the Pythagorean theorem. In this case, the cars are moving at right angles to each other— one goes east while the other goes north. After they've traveled for a while, the straight-line distance between the two cars will be the hypotenuse of the right triangle formed by their paths.
Let's define the variables as follows:
- The eastbound car has traveled 3 miles, which is one leg of the right triangle (let's call it "a").
- The northbound car has traveled an unknown distance, which is the other leg of the triangle (let's call it "b").
- The straight-line distance between the two cars is the hypotenuse of the right triangle and is given as 5 miles (let's call it "c").
The Pythagorean theorem states that:
a^2 + b^2 = c^2
We know that:
a = 3 miles
c = 5 miles
We can plug these values into the theorem and solve for b:
3^2 + b^2 = 5^2
9 + b^2 = 25
b^2 = 25 - 9
b^2 = 16
Take the square root of both sides to find b:
b = √16
b = 4
The northbound car has traveled 4 miles.
Hannah's bedroom is rectangular. The length of one wall of Hannah's bedroom is 16 feet. The length from one corner of the bedroom to the diagonally opposite corner is 20 feet. What is the length of the other wall?
Using the Pythagorean theorem again, we can solve for the length of the other wall in Hannah's bedroom. The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse (c) is equal to the sum of the squares of the lengths of the other two sides (a and b).
Where:
c = the diagonal across the room (20 feet),
a = the length of one wall (16 feet),
b = the length of the other wall, which we want to find.
The equation based on the theorem is:
a^2 + b^2 = c^2
Plugging the known values into the equation we have:
16^2 + b^2 = 20^2
256 + b^2 = 400
Now solve for b^2:
b^2 = 400 - 256
b^2 = 144
Take the square root of both sides to find b:
b = √144
b = 12
Therefore, the length of the other wall is 12 feet.
Dan is building a skateboarding ramp by propping the end of a piece of wood on a cinder block. If the ramp starts 72 centimeters from the cinder block and the piece of wood is 90 centimeters long, how tall is the cinder block?
Dan has effectively created a right-angled triangle with his skateboarding ramp, where the piece of wood acts as the hypotenuse and is 90 centimeters long. The distance from the ramp's base to the cinder block is one leg of the triangle and measures 72 centimeters. We need to find the height of the cinder block, which represents the other leg of the triangle.
We can use the Pythagorean theorem, which relates the sides of a right-angled triangle as follows:
a^2 + b^2 = c^2
In this scenario:
a = the height of the cinder block (which we are trying to find),
b = the horizontal distance from the ramp's base to the cinder block (72 cm),
c = the length of the ramp, i.e., the hypotenuse (90 cm).
Substituting the known values into the Pythagorean theorem, we get:
a^2 + 72^2 = 90^2
a^2 + 5184 = 8100
Now we solve for a^2:
a^2 = 8100 - 5184
a^2 = 2916
Finally, we find the value of a by taking the square root of both sides:
a = √2916
a = 54
Therefore, the height of the cinder block, which is the vertical distance from the ground to the top of the ramp where the wood is resting, is 54 centimeters.
Two airplanes leave the same airport. One heads north, and the other heads east. After some time, the northbound airplane has traveled 21 miles, and the eastbound airplane has traveled 72 miles. How far apart are the two airplanes?
The two airplanes are creating a right-angled triangle with their paths. The northbound airplane's path represents one leg of the triangle, and the eastbound airplane's path represents the other leg. The straight-line distance between the airplanes will be the hypotenuse of the right triangle.
Let's denote:
- The distance traveled by the northbound airplane as "a" equals 21 miles
- The distance traveled by the eastbound airplane as "b" equals 72 miles
- The straight-line distance between the airplanes as "c"
Using the Pythagorean theorem:
a^2 + b^2 = c^2
Plugging in the values for a and b, we can solve for c:
21^2 + 72^2 = c^2
441 + 5184 = c^2
5625 = c^2
Now, we find the square root of 5625 to solve for c:
c = √5625
c = 75
Thus, the two airplanes are 75 miles apart.