Question

1. a water pitcher has 118 ml of water in it. the pitcher is being filled at a constant rate at the kitchen sink. after 3 seconds, the water pitcher has 535 ml of water in it. write an equation for y, the volume of water in the pitcher, in ml, after x seconds.
2. the formula p = 6.31h + 41.5 can be used to predict the points p, earned by a student on an organic chemistry final, based on the number of hours h, that he or she studies. what is the meaning of the number 6.31 in context?
3. luke is driving up north to charlevoix. the table below shows the number of miles driven and the gallons of fuel in his car’s tank.

miles driven	50	100	150	200

assuming the gallons of fuel decrease at a constant rate during the trip, answer the following.
a. how many gallons of fuel are in his tank when he starts the trip?
b. how many gallons of fuel are used to drive 1 mile?

Explanation:

7.

Step1: Find the rate of filling

The pitcher starts with 118 mL and after 3 seconds has 535 mL. The amount of water added in 3 seconds is \(535 - 118=417\) mL. The rate of filling \(r\) (in mL per second) is \(\frac{417}{3}=139\) mL/s.

Step2: Write the linear - equation

The general form of a linear equation is \(y = mx + b\), where \(m\) is the slope (rate) and \(b\) is the y - intercept (initial amount). Here, \(m = 139\) and \(b = 118\). So the equation is \(y=139x + 118\).

The formula \(P = 6.31H+41.5\) is in the form of a linear equation \(y = mx + b\). In the context of predicting points on an organic chemistry final, the number 6.31 represents the increase in the number of points earned per hour of study.

Step1: Find the equation of the line

First, find the slope \(m\) of the line representing the relationship between miles driven (\(x\)) and gallons of fuel in the tank (\(y\)). Using two points \((x_1,y_1)=(50,12.7)\) and \((x_2,y_2)=(100,10.9)\), the slope \(m=\frac{y_2 - y_1}{x_2 - x_1}=\frac{10.9 - 12.7}{100 - 50}=\frac{- 1.8}{50}=-0.036\).
Using the point - slope form \(y - y_1=m(x - x_1)\) with \((x_1,y_1)=(50,12.7)\) and \(m=-0.036\), we get \(y-12.7=-0.036(x - 50)\), which simplifies to \(y=-0.036x+14.5\).

Step2a: Find the initial amount of fuel

When \(x = 0\) (starts the trip), \(y=-0.036\times0 + 14.5 = 14.5\) gallons.

Step2b: Find the fuel used per mile

The slope of the line represents the change in the number of gallons of fuel per mile driven. The slope \(m=-0.036\), so 0.036 gallons of fuel are used to drive 1 mile.

Answer:

\(y = 139x+118\)

8.