Question

1. students at tahayghen secondary school sell punch during the school carnival. the number of cups sold, n, is a linear function of the temperature in degress celsius, t. the students sold 458 cups when the temperature was 25°c. they sold 534 cups when the temperature was 29°c.

a) write an equation in slope - point form to represent this function
b) use the equation in part a to determine the approximate temperature when they sell 325 cups of punch.

Explanation:

Part (a)

Step 1: Identify two points

We have two points \((t_1, n_1)=(25, 458)\) and \((t_2, n_2)=(29, 534)\) where \(t\) is temperature and \(n\) is number of cups sold.

Step 2: Calculate the slope \(m\)

The formula for slope \(m=\frac{n_2 - n_1}{t_2 - t_1}\). Substituting values: \(m=\frac{534 - 458}{29 - 25}=\frac{76}{4} = 19\).

Step 3: Write slope - point form

Slope - point form is \(n - n_1=m(t - t_1)\) or \(n - n_2=m(t - t_2)\). Using the point \((29, 534)\) and \(m = 19\), the equation is \(n-534 = 19(t - 29)\). (We could also use the point \((25, 458)\) to get \(n - 458=19(t - 25)\))

Step 1: Substitute \(n = 325\) into the equation

Using the equation from part (a), say \(n-534=19(t - 29)\). Substitute \(n = 325\): \(325-534=19(t - 29)\)

Step 2: Simplify the left - hand side

\(325 - 534=-209\), so \(-209 = 19(t - 29)\)

Step 3: Solve for \(t\)

Divide both sides by 19: \(\frac{-209}{19}=t - 29\), \(-11=t - 29\). Then add 29 to both sides: \(t=-11 + 29=18\)

Answer:

\(n - 534=19(t - 29)\) (or \(n - 458 = 19(t - 25)\))

Part (b)