Question

there are four tanks of water. the amount of water in gallons, a, in tank a is given by the function a = 200 + 8t, where t is in minutes. the amount of water in gallons, b, in tank b starts at 400 gallons and is decreasing at 5 gallons per minute. these functions work when t ≥ 0 and t ≤ 80.

1. which tank started out with more water?
2. write an equation representing the relationship between b and t.
3. one tank is filling up. the other is draining out. which is which? how can you tell?
4. the amount of water in gallons, c, in tank c is given by the function c = 800 − 7t. is it filling up or draining out? can you tell just by looking at the equation?
5. the graph of the function for the amount of water in gallons, d, in tank d at time t is shown. is it filling up or draining out? how do you know? (with a graph of a line decreasing from d-intercept to t-intercept)

Explanation:

Step1: Find initial water for Tank A

To find the initial amount of water in Tank A, substitute \(t=0\) into \(A = 200 + 8t\):
\(A(0) = 200 + 8(0) = 200\) gallons

Step2: Compare initial water amounts

Tank B starts at 400 gallons. Compare 200 and 400:
\(400 > 200\)

Step3: Write Tank B's equation

Tank B has an initial value of 400 and decreases at 5 gallons per minute, so use the form \(B = \text{initial} - \text{rate} \times t\):
\(B = 400 - 5t\)

Step4: Identify filling/draining for A and B

For Tank A: the coefficient of \(t\) is positive (+8), so water increases. For Tank B: the coefficient of \(t\) is negative (-5), so water decreases.

Step5: Analyze Tank C's function

For \(C = 800 - 7t\), the coefficient of \(t\) is negative (-7), so water decreases over time.

Step6: Analyze Tank D's graph

The graph has \(D\) (water amount) decreasing as \(t\) (time) increases, so water is draining out.

Answer:

1. Tank B started out with more water.
2. \(B = 400 - 5t\)
3. Tank A is filling up (positive coefficient of \(t\) in its function), Tank B is draining out (negative coefficient of \(t\) in its function).
4. Tank C is draining out. The coefficient of \(t\) is negative (-7), which means the amount of water decreases as time increases.
5. Tank D is draining out. As time \(t\) increases, the amount of water \(D\) decreases, shown by the downward-sloping graph.