2.4 mil libs price sold 5.50 per lb amd 4.8 mil libs 4.50 ld. what linear function 1(p) expresses the amount

Let me restate this cryptic question.

2.4 million lb (pounds) of almonds are sold when the selling price is $5.50 per lb.

4.8 million lb (pounds) of almonds are sold when the selling price is $4.50 per lb.

What linear function expresses the amount of almonds sold as a function of the price per pound?

Answer:

Here selling price is the independent variable (x-axis) and almond sales in pound is a function of price (y-axis).

We can see that as price of almonds goes up, its sales decrease. So we expect a negative slope for this linear equation line.

Here we need to find the equation of line that passes through two points (x1, y1)=(5.5, 2.4) and (x2, y2)=(4.5, 4.8).

Solution using equation of line in 2 points form

(y-y1)/(y1-y2) = (x-x1)/(x1-x2) substitute co-ordinates of above two points in this equation

(y - 2.4)/(2.4 - 4.8) = (x - 5.5)/(5.5 - 4.5)

(y-2.4)/(-2.4) = (x - 5.5)/1

(y-2.4)/(-2.4) * -2.4 = (x - 5.5)/1 * -2.4 multiply both sides by -2.4

y - 2.4 = -2.4 (x - 5.5)

y - 2.4 = -2.4x + 13.2 ...remember when you multiply 2 negative numbers, you get a positive number

add 2.4 to both sides

y - 2.4 + 2.4 = -2.4x + 13.2 + 2.4

y = -2.4x + 15.6 (where x is price per pound of almonds and y is almonds sold in million pounds)

Alternative method by using slope-intercept form

Equation of line is given by

y = mx + c where m is the slope and c is the y-intercept

Slope of line passing through two points (x1, y1) and (x2, y2) is m = (y1-y2)/(x1-x2)

y = mx + c ....substitute above value of slope m.

y = (y1-y2)/(x1-x2)*x + c where c is the constant y-intercept

Substitute values of (x1, y1) and (x2, y2) in this equation

y= (2.4 - 4.8)/(5.5 - 4.5) *x +c

y=-2.4/1 *x +c y=-2.4x+c...................(1)

substitute (x1, y1) = (5.5, 2.4) in above equation to solve for c

2.4 = -2.4*5.5 +c

2.4 = -13.2 +c ....add 13.2 on both sides to get value of c

2.4 + 13.2 = -13.2 +c + 13.2

15.6 = c Substitute this value in equation (1) above to get

y = -2.4x + 15.6

Check:

We see that slope of the linear equation is negative. This is as expected.

substitute x = 5.5 in above equation and see if you get y = 2.4

substitute x = 4.5 in above equation and see if you get y = 4.8

The total value of the almonds sold should be:

(2.4 mil)(5.50 ea) + (4.8 mil)(4.50 ea) = 34.8 million dollars.

If we want a function of lbs of almonds sold as a function of price, we want the ratio (lbs sold)/(total price), so we want

(2.4 + 4.8 mil)/[(2.4 mil)(5.50 ea) + (4.8 mil)(4.50 ea)] = 0.2183908 lbs/$.