1) A firefighter aims a hose at a window 25 ft. above the ground. The equation
h = -0.01d^2 + 1.06d + 5
models the path of the water, when h equals height in feet. Estimate, to the nearest foot, the horizontal distance d in feet btween the firefighter and the building.
This problem confuses me a lot.
25 = -0.01d^2 + 1.06d + 5
so
.01 d^2 - 1.06 d + 20 = 0
d^2 - 106 d + 2000 = 0
d = (1/2)[ 106 +/- sqrt (106^2 - 8000)]
d = .5 [ 106 +/- sqrt (3236)]
d = .5 [ 106 +/- 57 ]
d = 81.5
or
d = 24.5
If he shoots the water up into the window, he is 25 feet away but if he shoots the water up so that it enters the window on the way back down he can be 82 feet away.
I understand that this problem might seem a bit confusing at first. However, let me break it down for you step by step.
First, let's understand the given equation: h = -0.01d^2 + 1.06d + 5
- In this equation, h represents the height of the water from the ground in feet.
- d represents the horizontal distance between the firefighter and the building in feet.
The goal is to estimate the horizontal distance (d) between the firefighter and the building when the water reaches a height of 25 ft.
To find this distance, we can substitute h = 25 into the equation and solve for d. Let's break it down further:
Step 1: Substitute h = 25 into the equation:
25 = -0.01d^2 + 1.06d + 5
Step 2: Rearrange the equation to form a quadratic equation:
-0.01d^2 + 1.06d + 5 - 25 = 0
Step 3: Simplify the equation:
-0.01d^2 + 1.06d - 20 = 0
Now, we can solve this quadratic equation to find the value(s) of d.
You have a few options to solve this equation:
- Factoring: Try to factor the quadratic equation into two binomials. However, this particular equation may not be easily factorable.
- Quadratic formula: You can use the quadratic formula to find the value(s) of d. The quadratic formula is given by:
d = (-b ± √(b^2 - 4ac)) / (2a)
In this equation, a = -0.01, b = 1.06, and c = -20.
By plugging these values into the quadratic formula and solving for d, you will find the horizontal distance between the firefighter and the building when the water reaches a height of 25 ft. Round your answer to the nearest foot.