Question

PLEASE HELP ME
How tall, in cm, is 1 cup? Explain how you determined the height of 1 cup?

Answer 1

Answer: 14 cm

=============================================

Step-by-step explanation:

x = number of cups

y = total height

We have these ordered pairs

(x1,y1) = (2,16)

(x2,y2) = (4,20)

Let's find the slope of the line through those points.

`(x_1,y_1) = (2,16) \text{ and } (x_2,y_2) = (4,20)\\\\m = \text{slope} = \frac{\text{rise}}{\text{run}} = \frac{\text{change in y}}{\text{change in x}}\\\\m = \frac{\text{y}_(2) - \text{y}_(1)}{\text{x}_(2) - \text{x}_(1)}\\\\m = (20 - 16)/(4 - 2)\\\\m = (4)/(2)\\\\m = 2\\\\`

A slope of 2, aka 2/1, means each time x goes up by 1, the y value goes up by 2.

It means each time we add a cup, the total height goes up by 2 cm.

This isn't the final answer, but it helps get us there.

------------

We'll then use point-slope form to determine the equation.

`y-y_1 = m(x - x_1)\\\\y-16 = 2(x - 2)\\\\y = 2(x - 2)+16\\\\y = 2x - 4+16\\\\y = 2x + 12\\\\`

The last step is to plug x = 1 into that equation to determine the height of 1 cup.

So,

`y = 2x + 12\\\\y = 2*1 + 12\\\\y = 2 + 12\\\\y = 14\\\\`

We have x = 1 cup lead to a total height of y = 14 cm.

The final answer is 14 cm.

-----------------

Notice that adding on 2 cm (the slope) gets us to 14+2 = 16 cm which is the height of two cups stacked together. A lot of the 2nd cup's height is part of the 1st cup's height. Only the 2 cm at the top is going to be added.

We could see that if we used x = 2.

`y = 2x + 12\\\\y = 2*2 + 12\\\\y = 4 + 12\\\\y = 16\\\\`

That confirms the left-most part of the diagram.

Now try x = 4.

`y = 2x + 12\\\\y = 2*4 + 12\\\\y = 8 + 12\\\\y = 20\\\\`

That works as well. It helps confirm we have the correct equation, and also confirms the final answer is correct. Another way to confirm is to either graph or use a table.

If you wanted to find the height of 8 cups, then,

`y = 2x + 12\\\\y = 2*8 + 12\\\\y = 16 + 12\\\\y = 28\\\\`