Question

You are buying dessert for your friend's birthday party. The Bake Shop sells cupcakes for $3 each and donuts for $2 each. Your budget is $42 for desserts.

Answer 1

Here, we are told to identify the number of donuts with d and the number of cupcakes with c

a) Firstly, we want to write an equation relating the number of donuts and the number of cupcakes that can be bought

1 donut costs $2, so the cost of d donuts will be d * 2 = $2d

1 cupcake costs $3 , so the cost of cupcakes will be c * 3 = $3c

Now, using the budget, we can write an equation for the number of cupcakes and desserts that can be bought

Mathematically, that will be;

`2d\text{ + 3c = 42}`

b) Now, given that c is 4, we want to calculate the value of d that will make the equation true

Mathematically, what we need to do here is to substitute c = 4 into the equation

We have;

`\begin{gathered} 2d\text{ + 3(4) = 42} \\ 2d\text{ + 12 = 42} \\ \\ 2d\text{ = 42-12} \\ \\ 2d\text{ = 30} \\ \\ d\text{ = }(30)/(2) \\ \\ d\text{ = 15} \end{gathered}`

c) What the above means in the context of the equation we have written is that the number of donuts we can buy alongsides 4 cupcakes is 15

So, buying 4 cupcakes, we can afford to still buy 15 donuts from the budget

d) Here, we want to have a graph of the equation

From the plot, we are told to make the number of donuts the y-axis and the number of cupcakes on the y-axis

Generally, the equation of a linear model can be represented as;

`\begin{gathered} y\text{ = mx + b} \\ \\ m\text{ is slope and b is y-intercept} \\ \end{gathered}`

Now, let us replace d with y and c with x; we have the result as;

`\begin{gathered} d\text{ = cx + b} \\ \\ \text{Let us write the equation we have in this form} \\ \\ 2d\text{ + 3c = 42} \\ \\ 2d\text{ = -3c + 42} \\ \\ d\text{ = }(-3c)/(2)\text{ + 21} \end{gathered}`

So for our y-intercept ( the point at which the line touches the y-axis), we have the coordinate of the point as (21,0)

To get for ths x-intercept, replace d with 0;

`\begin{gathered} (3c)/(2)\text{ = 21} \\ \\ 3c\text{ = 21 }*\text{ 2} \\ \\ 3c\text{ = 42} \\ \\ c\text{ = }(42)/(3) \\ \\ c\text{ = 14} \end{gathered}`

S0, to graph the line, we simply join (21,0) on the y-axis with (14,0) on the x-axis

A plot representing this is given below;