My answer doesn't make sense. I will show question then my work: A city holding tank for water sits 20 m above the city. If a house was on fire 15 m
away from the base of the holding tank, and the tank was originally filled to a depth of
15 m, to what depth would firefighters have to drain the tank so water, sprayed from
the bottom of the tank, hits the bottom of the house?
step1: Viy=0, a=g=9.8m/s/s, dy=20
dy=Viyt + 1/2 at^2 -->
t = sqrt(40m/9.8m/s/s)= 2.02s
Step2: Vx=dx/t and Vx=15m/2.02s=7.43m/s
Step 3: change in x = 15m (distance house from water tower) and change in h (depth of water) = 15m
change in X = Vx(sqrt(2h/g) or
h = [(x^2)(g)]/[(Vx^2)(2)]
h = ((15^2)(9.8)]/[(7.43^2)(2)] m
h= 19.97m
That's more meters than I have. Not sure what I missed.
Well, it seems like you took all the right steps in your calculations. However, sometimes math can be a bit tricky, just like solving a Rubik's cube while riding a unicycle.
In this case, I believe the issue lies in rounding. Rounding errors can be quite pesky little things, like a clown squeezing into a tiny car. When you rounded the value of h to two decimal places, you got 19.97m, which is just a whisker away from 20m. It's like trying to stretch your arm to reach the last slice of pizza but coming up just short.
So, in this scenario, if we take into account those sneaky rounding errors and round the value of h up to two decimal places, we get 20.00m. Ta-da! Now it all makes sense, like a joke that finally gets a laugh.
Remember, math can be a bit of a jester at times, but with patience and a sense of humor, you can solve any problem, whether it's calculating depths or dodging banana peels.
Based on your work, it seems that there is a minor calculation error in Step 3. Let me provide the correct step-by-step calculation for you:
Step 1: Calculate the time it takes for the water to reach the ground.
Using the equation: dy = Viy * t + (1/2) * a * t^2
Given: Viy = 0 m/s (initial vertical velocity), a = g = 9.8 m/s^2 (acceleration due to gravity), and dy = 20 m (initial height of the tank)
20 = 0 * t + (1/2) * 9.8 * t^2
20 = 4.9 * t^2
t^2 = 20 / 4.9
t = sqrt(20 / 4.9)
t ≈ 2.02 s
Step 2: Calculate the horizontal velocity of the water.
Using the equation: Vx = dx / t
Given: dx = 15 m (distance of the house from the base of the tank) and t ≈ 2.02 s
Vx = 15 / 2.02
Vx ≈ 7.43 m/s
Step 3: Calculate the depth of the water needed to hit the bottom of the house.
Using the equation: h = (dx^2 * g) / (2 * Vx^2)
Given: dx = 15 m (distance of the house from the base of the tank) and Vx ≈ 7.43 m/s
h = (15^2 * 9.8) / (2 * 7.43^2)
h ≈ (225 * 9.8) / (2 * 54.9649)
h ≈ 2205 / 109.9298
h ≈ 20.05 m
Therefore, the correct depth to which the firefighters would have to drain the tank so that water sprayed from the bottom of the tank hits the bottom of the house is approximately 20.05 meters.
Let's go through your work step by step to check for any errors.
Step 1: You correctly used the equation dy = Viyt + 1/2 at^2 to find the time it takes for water to fall from the tank. The calculated time t = sqrt(40m/9.8m/s^2) is approximately 2.02s.
Step 2: You correctly found the horizontal velocity Vx using the formula Vx = dx/t. The calculated Vx = 15m/2.02s is approximately 7.43m/s.
Now let's analyze Step 3, where it seems you might have made a mistake.
The equation you used, change in x = Vx(sqrt(2h/g)), is incorrect. The correct equation should be derived from the kinematic equation for horizontal motion:
dx = Vx * t
Since you've already calculated the value of Vx and t, you can directly substitute these values into the equation:
change in x = Vx * t = (15m/2.02s) * 2.02s = 15m
This means that the horizontal distance between the base of the tank and the house (15m) remains the same.
To find the new depth of water required (h), you can rearrange the equation:
dx = Vx * sqrt(2h/g)
Solving for h:
h = (dx^2 * g) / (Vx^2 * 2)
Plugging in the values:
h = [(15m)^2 * 9.8m/s^2] / [(7.43m/s)^2 * 2] ≈ 14.8m
So, the correct answer is that firefighters would have to drain the tank to a depth of approximately 14.8 meters for water sprayed from the bottom of the tank to hit the bottom of the house.