Linear Equation Help

Question

Margaret purchased a new bar of soap. Three days after she originally used the soap, she was curious how much soap per day she was using. She decided to weigh her soap and found that the bar was 103 grams. Four days later she re-measured the same bar of soap and recorded a weight of 80 grams. Assuming that Margaret uses the same amount of soap daily (and that she used the soap daily), write a function S that shows the amount of soap remaining after d days of use.

Answer 1

Given: We are given data for weight of the bar of soap (assign to y) and the number of days (assign to x). Two points: (x1, y1) = (0, 123) and (x2, y2) = (4, 80)

Plan: Use the slope formula to find the slope. Then use the "Point Slope" formula to find the equation of the line.

Slope =>

m = (y2 -y1) / (x2 - x1) // substitue in (x1, y1) and (x2, y2)

m = (123 - 80) / (0 - 4)

m = - 23/4

Point Slope =>

(y - y1) = m(x -x1) // substitue in (x1, y1) and m

(y - 123) = (-23/4)(x - 0)

y = -23/4x + 123

Answer 2

This is a question about finding a function that relates the day to the amount of soap REMAINING.

So on the first day there was some amount of soap, call it S(0).

We know that on day 3 she weighed her soap and there were 103 grams remaining. We can write this as S(3) = 103

Similarly, on day 7 she weighed her soap and found 80 grams remaining S(7) = 80.

The first step I recommend is finding the rate of soap consumption, because we're told that is constant.

So, we have 2 data points and we want to find SOAP per DAY, so we need to find the amount of soap used in a certain number of days, then divide by the number of days.

soap per day = (103-80)/4 = 23/4 = 5.75 grams of soap per day

Technically it's -5.75 grams of soap per day because we're "losing" total soap

Now that we have the rate of soap usage we can write the function S(d) as S(d) = -5.75d+S(0), where S(0) is the amount of soap at day 0.

To find S(0) we use the constant rate of soap consumption (5.75 per day) and find how much we had 3 days before we measured on day 3. Over those 3 days we would have used 5.75*3 or 17.25g of soap, so S(0) = S(3) + 17.25 = 103+17.25 = 120.25

So, our equation S(d) should be S(d) = -5.75*d+120.25