A wire is stretched from the ground to the top of an antenna tower. The wire is 15 feet long. The height of the tower is 3 feet greater than the distance d from the tower's base to the end of the wire. Find the distance d and the height of the tower.
Oh, I've heard of this wire-high-tower situation before! Let's do some clown calculations!
Let's call the distance from the tower's base to the end of the wire "d." According to the problem, the wire itself is 15 feet long. So, if we subtract the height of the tower from the length of the wire, we'll get the value of "d."
Let's say the height of the tower is "h." They say the height of the tower is 3 feet greater than "d." That means we can write this equation: h = d + 3.
Now, if we add "d" and "h" together, we get the length of the wire: d + h = 15.
Since we already have an equation for "h," we can substitute it into the second equation: d + (d + 3) = 15.
Now we can solve it like true circus mathematicians: 2d + 3 = 15.
Subtracting 3 from both sides, we find that 2d = 12.
Finally, we divide both sides by 2, and we find that d = 6. So, the distance from the tower's base to the end of the wire is 6 feet.
Remember, the height of the tower is 3 feet greater than d, so the height is d + 3, which is 6 + 3 = 9 feet.
So, the distance "d" is 6 feet, and the height of the tower is 9 feet.
Let's assume the distance d from the tower's base to the end of the wire is x.
According to the given information, the height of the tower is 3 feet greater than d. So the height of the tower is x + 3.
We also know that the wire is stretched from the ground to the top of the tower, making the wire's length equal to the height of the tower. Therefore, the wire's length is also x + 3.
Since the wire's length is given as 15 feet, we can set up the following equation:
x + 3 = 15
Simplifying the equation:
x = 15 - 3
x = 12
So the distance d from the tower's base to the end of the wire is 12 feet.
Substituting this value into the equation for the height of the tower:
height = d + 3 = 12 + 3 = 15
Therefore, the height of the tower is 15 feet and the distance d is 12 feet.
To solve this problem, we can use the Pythagorean theorem, which states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.
Let's assume that "d" represents the distance from the tower's base to the end of the wire. According to the problem, the height of the tower is 3 feet greater than "d".
We can set up the equation as follows:
d^2 + (d + 3)^2 = 15^2
First, we square both sides:
d^2 + (d + 3)^2 = 225
Expanding the equation:
d^2 + (d^2 + 6d + 9) = 225
Combining like terms:
2d^2 + 6d + 9 = 225
Moving all terms to one side of the equation:
2d^2 + 6d + 9 - 225 = 0
Simplifying:
2d^2 + 6d - 216 = 0
Now, we can solve this quadratic equation by factoring, completing the square, or using the quadratic formula. In this case, let's use factoring:
2d^2 + 6d - 216 = 0
Dividing through by 2:
d^2 + 3d - 108 = 0
Now, we need to factor this quadratic equation:
(d + 12)(d - 9) = 0
Setting each factor to zero:
d + 12 = 0 or d - 9 = 0
Solving for "d":
d = -12 or d = 9
Since distance cannot be negative in this context, we discard the negative value.
Therefore, the distance "d" is 9 feet.
Now, we can find the height of the tower by substituting the value of "d" into the equation:
Height = d + 3
= 9 + 3
= 12 feet
So, the distance "d" is 9 feet and the height of the tower is 12 feet.
by Pythagoras ... (d + 3)^2 + d^2 = 15^2
2 d^2 + 6 d - 216 = 0 ... d^2 + 3 d - 108 = 0
factoring ... (d + 12)(d - 9) = 0