Moderate# Soap Suds

GMAT-NOELQQ

A manufacturer of dish detergent believes the height of soapsuds in the dishpan would increase if the amount of detergent used in washing dishes were increased. To test this claim, eleven pans of water were prepared. All pans were the same size and contained the same amount of water at the same temperature. Eleven different amounts of dish detergent (in grams) were added to each pan.

The water in the dishpan was agitated for a set amount of time, and the height of the resulting suds (in centimeters) was measured. The data is graphed in the scatterplot below, with two linear models shown to fit the data. The amount of detergent is on the horizontal axis and the height of the soap suds is on the vertical axis.

If $y$ represents the suds height and $x$ represents the amount of detergent, the first linear model to fit the pattern of data has the equation $y = 0.677x + 0.65$. The second linear model has the equation $y = 0.823x$. The second model has a $y$-intercept of zero, since if there are zero grams of detergent added, the suds height should also be zero.

**Part 1:**

If the amount of detergent in one typical dishpan were increased from $6.0$ grams to $7.2$ grams,

centimeters is the difference in the increases in soap suds heights predicted by the two linear models.

**Part 2:**

The two models predict the same suds height for an amount of detergent of

grams.

**Part 3:**

Suds of height $7$ centimeters can be achieved with the second model with

grams less detergent than with the first model.