Devon's bike has wheels that are 27 inches in diameter. After the front wheel picks up a tack, he rolls another 100 ft and stops. How far above the ground is the tack?
To determine how far above the ground the tack is, we can use the concept of similar triangles. Here's how:
1. Start by drawing a diagram to represent the situation. Draw a triangle to represent Devon's bike resting on the ground, with the front and back wheels labeled.
- Label the distance from the front wheel to the tack as x.
- Label the distance from the back wheel to the tack as y.
- Label the distance between the two wheels as d (which we'll calculate later).
- Draw a vertical line from the tack to the ground.
2. We can form similar triangles by looking at the triangle formed by the front wheel, the tack, and the vertical line to the ground. The other triangle is formed by the back wheel, the tack, and the vertical line.
3. The ratio of the corresponding sides in the similar triangles is the same.
- In the first triangle, the length of the side opposite the tack is x.
- In the second triangle, the length of the side opposite the tack is (y + 100) feet (since Devon rolled another 100 ft after the tack).
4. Since the triangles are similar, we can set up the following proportion:
- x / (y + 100) = 27 inches (since the diameter of the wheel is 27 inches).
5. Now, let's solve for x. Cross-multiply the proportion:
- x * 27 = (y + 100).
- x = (y + 100) / 27.
6. To find the value of y (the height of the tack above the ground), we need to find the distance between the front and back wheels of the bike, d. We can use the Pythagorean theorem:
- d^2 = x^2 + y^2.
7. Plugging in the value of x from step 5, we get:
- d^2 = ((y + 100) / 27)^2 + y^2.
8. We also know that the distance between the front and back wheels is the sum of the distances from each wheel to the tack:
- d = x + y.
9. Now, we have two equations:
- d^2 = ((y + 100) / 27)^2 + y^2,
- d = x + y.
10. Substitute the value of x from step 5 into the second equation:
- d = (y + 100) / 27 + y.
11. Simplify the equation:
- d = (2y + 100) / 27.
12. Now we can substitute the value of d from step 11 into the first equation:
- ((2y + 100) / 27)^2 = ((y + 100) / 27)^2 + y^2.
13. Solve the equation using algebra. Simplify both sides of the equation and isolate y.
This process will lead us to the solution and tell us how far above the ground the tack is.
To determine the distance above the ground where the tack is located, we can use the concept of similar triangles.
Given that the diameter of the wheel is 27 inches, the radius would be half of the diameter, which is 27/2 = 13.5 inches.
Let's assume that the height of the tack from the ground is h inches.
Using the similar triangles concept, we can set up the proportion:
(Radius of the wheel) / (Distance rolled) = (Distance from the tack to the ground) / (Distance rolled + Distance from the tack to the ground)
Substituting the given values, we get:
13.5 inches / 100 ft = h / (100 ft + h inches)
To calculate in the same unit, we need to convert 100 ft to inches:
100 ft * 12 inches/ft = 1200 inches
Rewriting the equation, we have:
13.5 / 1200 = h / (1200 + h)
Solving this equation for h, we can cross-multiply:
13.5 * (1200 + h) = 1200 * h
16200 + 13.5h = 1200h
1200h - 13.5h = 16200
1186.5h = 16200
h = 16200 / 1186.5
h ≈ 13.66 inches
Therefore, the tack is located approximately 13.66 inches above the ground.