Question

A tank holds 50 gal of water, which drains from a leak at the bottom, causing the tank to empty in 20 min. The tank drains faster when it is nearly full because the pressure on the leak is greater. Torricelli's Law gives the volume of water remaining in the tank after t minutes as V(t)=50(1−t20)20≤t≤20 (a) Find V(0) and V(20). (b) What do your answers to part (a) represent? (c) Make a table of values of V(t) for t = 0, 5,10, 15, 20. (d) Find the net change in the volume V as t changes from 0 min to 20 min.

Answer 1

Answer:

(a) V(0) = 50 gal, V(20) = 0 gal

(b)At t= 0 the tank is full.

At t=0 the tank is empty

(c)

Time       volume

 0               50 gal

 5                37.5 gal

 10              25 gal

 15             12.5 gal

20                0 gal

(d)

Net change of volume = 50 gal

Step-by-step explanation:

Given that the capacity of the tank is 50 gal.

Torricelli's Law gives the volume of water remaining in the tank after t minutes as

$V(t)=50(1-\frac{t}{20})^2$

(a)

To find V(0), we put t = 0 in the above equation

$V(0)=50(1-\frac{0}{20})^2$

        $=50(1-0)^2$

        = 50 gal

To find V(20), we put t =2 0 in the above equation

$V(20)=50(1-\frac{20}{20})^2$

        $=50(1-1)^2$

        = 0 gal

(b)

At t= 0 the tank is full.

At t=0 the tank is empty.

(c)

Time                                          V(t)

  0                                  $50(1-\frac{0}{20})^2=50 \ gal$

  5                                  $50(1-\frac{5}{20})^2=37.5 \ gal$

 10                                  $50(1-\frac{10}{20})^2=25 \ gal$

 15                                 $50(1-\frac{15}{20})^2=12.5 \ gal$

 20                                $50(1-\frac{20}{20})^2=0$

(d)

Net change of volume = V(0) -V(20)

                                     =(50-0) gal

                                    = 50 gal

Answer 2

The net change in the volume V as t changes from 0 min to 20 min is 1000

The volume of the tank after t minutes is given as:

V(t)=50(20 - t), 0≤t≤20

(a) Find V(0) and V(20).

We have:

V(t)=50(20 - t)

Substitute 0 for t

V(0)=50(20 - 0) = 1000

Substitute 0 for t

V(20)=50(20 - 20) = 0

Hence, the values of V(0) and V(20) are 1000 and 0, respectively

(b) What do your answers to part (a) represent?

The answers represent the initial and the final volumes of the tank, respectively

(c) A table of values of V(t) for t = 0, 5,10, 15, 20.

We have:

V(0)=50(20 - 0) = 1000

V(5)=50(20 - 5) = 750

V(10)=50(20 - 10) = 500

V(15)=50(20 - 15) = 250

V(20)=50(20 - 20) = 0

So, the table of values is:

t           V(t)

0            1000

5             750

10            500

15             250

20            0

(d) The net change in the volume V as t changes from 0 min to 20 min.

This is the difference between V(20) and V(0).

So, we have:

Net = V(0) - V(20)

Net = 1000 - 0

Net = 1000

Hence, the net change in the volume V as t changes from 0 min to 20 min is 1000

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