A stone is dropped from a height of 100 feet. The time it takes for the stone to reach a height of h is given by the function t = 1/4 √(100 - h) , where t is the time in seconds. Identify the domain and range of the function, and determine the height of the stone after 2 seconds.
My work -
2 = 1/4 √(100 - h)
I multiplied both sides by 4 from -
8 = √(100 - h)
Then, I got -
8 = 10√(-h)
I divided by 10 -
8/10 = √(-h)
now what?
Do I divide by -1 on both sides?
My work -
2 = 1/4 √(100 - h)
I multiplied both sides by 4 from -
8 = √(100 - h)
------------- OK SO FAR BUT
square both sides
64 = 100 - h
h = 100-64
h = 36
domain from 100-h must be +
0<h<100
for range
if h = 0, t = 10/4 hits ground
if h = 100, t = 0 drop it then
so 0<=t<=10/4
============================
now let me check original
fell distance 100 - h = d
v = 32 t
d = 16t^2
t^2 = d/16
t = (1/4) sqrt d
t = (1/4) sqrt(100-h)
ok, valid
I am so dumb how did I not think of squaring.
Thanks.
You are welcome.
No, you don't need to divide by -1 in this case. Let's go back and check your calculations.
Starting from your equation:
8 = √(100 - h)
To isolate h, you need to square both sides of the equation:
(8)^2 = (√(100 - h))^2
This simplifies to:
64 = 100 - h
Next, we can isolate h by subtracting 100 from both sides:
64 - 100 = -h
Simplifying further, we get:
-36 = -h
To solve for h, multiply both sides by -1:
36 = h
Therefore, the height of the stone after 2 seconds is 36 feet.
Now, let's address the domain and range of the function. The domain refers to the valid input values for the function, in this case, the height (h). Since the stone is dropped from a height of 100 feet, the height cannot exceed 100. Therefore, the domain includes all values less than or equal to 100.
The range, on the other hand, refers to the valid output values for the function, in this case, the time (t). Looking at the function, notice that the time cannot be negative because it represents the time it takes for the stone to reach a certain height. So the range includes all positive values for t.
In summary:
Domain: h ≤ 100
Range: t > 0