Question

A tank holds 3000 gallons of water, which drains from the bottom of the tank in half an hour. The values in the table show the volume V of water remaining in the tank (in gallons) after t minutes.
t (min) 5 10 15 20 25 30
V (gal) 2097 1308 780 303 87 0
(a) If P is the point (15, 780) on the graph of V, find the slopes of the secant lines PQ when Q is the point on the graph with the following values. (Round your answers to one decimal place.)

Q slope
(5, 2097)
(10, 1308)
(20, 303)
(25, 87)
(30, 0)

Answer 1

Answer:

The slope of the secant line PQ is the following for each point:

Q                         Slope (a)

(5, 2097)              -131.7 gal/min

(10, 1308)              -105.6 gal/min

(20, 303)               -95.4 gal/min

(25, 87)                  -69.3 gal/min

(30, 0)                   -52 gal/min

Step-by-step explanation:

The slope can be easily obtained by using a linear approximation for the 2 points P and Q. In this case, with the slope a:

$f(t)=V(t)=at+b$

$f(t_P)=V_P=at_P+b$             (1)

$f(t_Q)=V_Q=at_Q+b$           (2)

if equations 1 and 2 are subtracted:

$V_P-V_Q=at_P -at_Q=a(t_P -t_Q)\\a=\frac{V_P-V_Q}{t_P -t_Q}$