A wire is stretched from the gound to the top of an antenna tower. The wire is 25 feet long. The height of the tower is 5ft greater than the distance d from the tower's base to the end of the wire. Find the distance and the height of the tower.
Let's solve this problem step-by-step.
Step 1: Define the variables.
Let d represent the distance from the tower's base to the end of the wire.
Let h represent the height of the tower.
Step 2: Write the given information as equations.
The wire is 25 feet long, so we have the equation: d + h = 25.
The height of the tower is 5 feet greater than the distance d, so we have the equation: h = d + 5.
Step 3: Solve the system of equations.
Substitute the second equation into the first equation to eliminate the variable h:
d + (d + 5) = 25.
Combine like terms:
2d + 5 = 25.
Subtract 5 from both sides:
2d = 20.
Divide both sides by 2:
d = 10.
Now substitute the value of d back into the second equation to find h:
h = 10 + 5.
h = 15.
Step 4: Check your answers.
Does d + h equal 25?
10 + 15 = 25.
Yes, it does.
Step 5: Answer the question.
The distance from the tower's base to the end of the wire is 10 feet, and the height of the tower is 15 feet.
To find the distance (d) and the height of the tower, we can set up a system of equations based on the given information.
Let's denote the distance from the base of the tower to the end of the wire as "d". According to the question, the height of the tower is 5 feet greater than this distance.
We can set up the first equation based on the length of the wire:
d + height of the tower = 25
Now, let's replace the "height of the tower" with "d + 5" (since the height is 5 feet greater than d):
d + (d + 5) = 25
Simplifying the equation, we have:
2d + 5 = 25
Subtracting 5 from both sides of the equation:
2d = 25 - 5
2d = 20
Dividing both sides of the equation by 2:
d = 20/2
d = 10
So, the distance from the base of the tower to the end of the wire (d) is 10 feet.
Now, let's find the height of the tower:
Height of the tower = d + 5
Height of the tower = 10 + 5
Height of the tower = 15
Therefore, the distance from the base of the tower to the end of the wire is 10 feet, and the height of the tower is 15 feet.
better yet, consider a scaled-up 3-4-5 right triangle.
Scale it up by a factor of 5 and you have 15-20-25 which fits the bill.