Math  /  Algebra

QuestionASKYOUR TEACHER V(t)=75(1t32)20t32V(t)=75\left(1-\frac{t}{32}\right)^{2} \quad 0 \leq t \leq 32. (a) Find V(0)V(0) and V(32)V(32). V(0)=galV(32)=gal\begin{array}{l} V(0)=\square \mathrm{gal} \\ V(32)=\square \mathrm{gal} \end{array} (b) What do your answers to part (a) represent? V(32)V(32) represents the the time when the tank is empty, and V(0)V(0) represents the time when it is full. V(0)V(0) represents the time when the tank is empty, and V(32)V(32) represents the time when it is full. V(0)V(0) represents the initial rate at which the water is leaking, and V(32)V(32) represents the rate at which it is leaking when 32 gallons have drained. V(0)V(0) represents the initial volume, and V(32)V(32) represents the final volume. V(32)V(32) represents the initial volume, and V(0)V(0) represents the final volume. (c) Make a table of values of V(t)V(t) for t=0,8,16,24,32t=0,8,16,24,32. (Round your answer to three decimal places.) \begin{tabular}{|c|c|} \hlinett (in minutes) & V(t)V(t) (in gallons) \\ \hline 0 & \square \\ 8 & \square \\ 16 & \square \\ 24 & \\ 32 & \\ \hline \end{tabular} (d) Find the net change in the volume VV as tt changes from 0 min to 32 min .

Studdy Solution

STEP 1

What is this asking? We're looking at a formula, V(t)V(t), that tells us how much water is in a tank at different times, and we want to understand what's happening to the water level over time. Watch out! Make sure to plug in the right values for tt and carefully calculate the results!
Also, don't mix up what V(0)V(0) and V(32)V(32) represent.

STEP 2

1. Calculate V(0)V(0) and V(32)V(32).
2. Interpret V(0)V(0) and V(32)V(32).
3. Create a table of values for V(t)V(t).
4. Calculate the net change in volume.

STEP 3

We're given the formula V(t)=75(1t32)2V(t) = 75\left(1 - \frac{t}{32}\right)^2.
To find V(0)V(0), we **substitute** t=0t = 0 into the formula: V(0)=75(1032)2V(0) = 75\left(1 - \frac{0}{32}\right)^2 V(0)=75(10)2V(0) = 75\left(1 - 0\right)^2V(0)=7512V(0) = 75\cdot 1^2V(0)=751V(0) = 75 \cdot 1V(0)=75V(0) = 75So, V(0)=75V(0) = \textbf{75} gallons.

STEP 4

Now, let's find V(32)V(32) by **substituting** t=32t = 32 into the formula: V(32)=75(13232)2V(32) = 75\left(1 - \frac{32}{32}\right)^2 V(32)=75(11)2V(32) = 75\left(1 - 1\right)^2V(32)=7502V(32) = 75\cdot 0^2V(32)=750V(32) = 75 \cdot 0V(32)=0V(32) = 0So, V(32)=0V(32) = \textbf{0} gallons.

STEP 5

V(0)=75V(0) = 75 gallons represents the **initial volume** of water in the tank at time t=0t = 0 minutes.

STEP 6

V(32)=0V(32) = 0 gallons represents the **final volume** of water in the tank at time t=32t = 32 minutes.
This means the tank is empty at t=32t = 32 minutes.

STEP 7

We already found V(0)V(0) and V(32)V(32).
Now, let's calculate V(8)V(8), V(16)V(16), and V(24)V(24).
For t=8t = 8: V(8)=75(1832)2=75(34)2=7591642.188V(8) = 75\left(1 - \frac{8}{32}\right)^2 = 75\left(\frac{3}{4}\right)^2 = 75 \cdot \frac{9}{16} \approx 42.188 gallons.
For t=16t = 16: V(16)=75(11632)2=75(12)2=7514=18.75V(16) = 75\left(1 - \frac{16}{32}\right)^2 = 75\left(\frac{1}{2}\right)^2 = 75 \cdot \frac{1}{4} = 18.75 gallons.
For t=24t = 24: V(24)=75(12432)2=75(14)2=751164.688V(24) = 75\left(1 - \frac{24}{32}\right)^2 = 75\left(\frac{1}{4}\right)^2 = 75 \cdot \frac{1}{16} \approx 4.688 gallons.

STEP 8

| tt (in minutes) | V(t)V(t) (in gallons) | |---|---| | 0 | 75 | | 8 | 42.188 | | 16 | 18.75 | | 24 | 4.688 | | 32 | 0 |

STEP 9

The net change in volume is the **difference** between the final volume and the initial volume: V(32)V(0)=075=75V(32) - V(0) = 0 - 75 = -75 gallons.

STEP 10

(a) V(0)=75V(0) = 75 gallons and V(32)=0V(32) = 0 gallons. (b) V(0)V(0) represents the initial volume, and V(32)V(32) represents the final volume. (c) See the table in step 2.3.2. (d) The net change in volume is 75-75 gallons.

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