Math  /  Algebra

QuestionA sink is filled with 54 quarts of water. The function q=f(t)=542.5tq=f(t)=54-2.5 t gives the volume of wate in the sink, in quarts, tt seconds after removing the stopper. a. Find f1f^{-1}. b. What does the input of f1f^{-1} represent? What does the output of f1f^{-1} represent? c. How long will it take for the sink to drain completely? d. How long will it take for the sink to lose one-third of its water?

Studdy Solution

STEP 1

What is this asking? We're looking at how the amount of water in a sink changes over time as it drains, and we want to figure out how long it takes for it to empty completely and to lose one-third of its water.
We also need to find the inverse function and explain what it represents. Watch out! Don't mix up the input and output of the function and its inverse!
Also, make sure to use the correct units (quarts and seconds).

STEP 2

1. Find the inverse function.
2. Interpret the inverse function.
3. Calculate the time to drain completely.
4. Calculate the time to lose one-third of the water.

STEP 3

We're given the function q=f(t)=542.5tq = f(t) = 54 - 2.5t, where qq is the volume of water in quarts and tt is the time in seconds.

STEP 4

To find the inverse, we **swap** qq and tt: t=542.5qt = 54 - 2.5q.

STEP 5

Now, we **solve for** qq.
First, we subtract 54 from both sides: t54=2.5qt - 54 = -2.5q.
Then, we divide both sides by -2.5: t542.5=q\frac{t - 54}{-2.5} = q.
So, q=f1(t)=54t2.5q = f^{-1}(t) = \frac{54 - t}{2.5}.

STEP 6

The input tt of f1(t)f^{-1}(t) represents the **volume of water** in the sink (in quarts).

STEP 7

The output of f1(t)f^{-1}(t) represents the **time** (in seconds) it takes to reach that volume of water.

STEP 8

When the sink is completely drained, the volume of water is **zero**: q=0q = 0.

STEP 9

We use the original function q=f(t)=542.5tq = f(t) = 54 - 2.5t and substitute q=0q = 0: 0=542.5t0 = 54 - 2.5t.

STEP 10

Now, we **solve for** tt: 2.5t=542.5t = 54, so t=542.5=21.6t = \frac{54}{2.5} = 21.6 seconds.

STEP 11

One-third of the initial volume is 1354=18\frac{1}{3} \cdot 54 = 18 quarts.

STEP 12

If the sink loses 18 quarts, the remaining water is 5418=3654 - 18 = 36 quarts.
So, we set q=36q = 36.

STEP 13

We substitute q=36q = 36 into q=542.5tq = 54 - 2.5t: 36=542.5t36 = 54 - 2.5t.

STEP 14

Now, we **solve for** tt: 2.5t=5436=182.5t = 54 - 36 = 18, so t=182.5=7.2t = \frac{18}{2.5} = 7.2 seconds.

STEP 15

a. f1(t)=54t2.5f^{-1}(t) = \frac{54 - t}{2.5} b. The input of f1(t)f^{-1}(t) represents the volume of water in the sink (in quarts), and the output represents the time (in seconds) it takes to reach that volume. c. It will take **21.6 seconds** for the sink to drain completely. d. It will take **7.2 seconds** for the sink to lose one-third of its water.

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