Question

pensions and uses
finding the initial amount and rate of change given a table for a linear...
a factory workers are draining a large vat containing water. the water is being drained at a constant rate. the table below shows
the amount of water in the vat after different amounts of time.
time (minutes) 15 20 25 30
water (liters) 620 560 500 440
answer the following questions.
(a) how much water was in the vat when the workers started draining it?
liters
(b) choose the statement that best describes how the time and the amount of
water in the vat are related. then give the value requested.
as time increases, the amount of water in the vat decreases.
at what rate is the amount of water in the vat decreasing?
liters per minute
as time increases, the amount of water in the vat increases.
at what rate is the amount of water in the vat increasing?
liters per minute

Explanation:

Part (a)

Step1: Find the rate of change (slope)

We use two points from the table, say \((15, 620)\) and \((20, 560)\). The slope \(m\) is calculated as \(m=\frac{y_2 - y_1}{x_2 - x_1}=\frac{560 - 620}{20 - 15}=\frac{- 60}{5}=- 12\). This means the water is decreasing at 12 liters per minute.

Step2: Use the point - slope form to find the initial amount

The point - slope form of a line is \(y - y_1=m(x - x_1)\). We can use the point \((15,620)\) and \(m=-12\). We want to find the value of \(y\) when \(x = 0\) (initial time).
\(y-620=-12(x - 15)\)
When \(x = 0\):
\(y-620=-12(0 - 15)\)
\(y-620=-12\times(-15)\)
\(y - 620 = 180\)
\(y=620 + 180=740\)

Part (b)

Step1: Analyze the relationship between time and water

Looking at the table, as the time (in minutes) increases from 15 to 20 to 25 to 30, the amount of water (in liters) decreases from 620 to 560 to 500 to 440. So the relationship is that as time increases, the amount of water in the vat decreases.

Step2: Calculate the rate of decrease

We use the formula for the rate of change (slope) between two points. Let's take \((15,620)\) and \((20,560)\). The rate of change \(r=\frac{560 - 620}{20 - 15}=\frac{-60}{5}=- 12\). The negative sign indicates a decrease, and the magnitude 12 is the rate at which the water is decreasing (12 liters per minute).

Answer:

740