SAT Mock Exam 2 — Math, No-Calculator Section: Solution to Question 1
In this question, we are given a mathematical linear expression for the payment that has to be made to a painter for painting walls of a building. We are told that the painter uses a specific brand of paint to paint walls of a building that are equal in size. The lenghts and heights of those walls, measured in feet, are given by h and l while the number of walls to be painted is given by n. The mathematical expression for the painter’s fee is given as:
Fee ($) = nKlh
Where K is a constant with units $/feet²
We know that for a given building, the area of the wall that the painter has to paint remains fixed and the price of the paint used also remain constants as the painter does not alternate paint from one wall to another. We are now being asked to find which variable (n, K, l, h) in the fee expression will be changed if the painter uses a more expensive brand of paint.
We already know that each factor is directly proportional to the payment that has to be made by the service offered by the painter. In other words, the more walls need to be painted (n), the higher the fee will be. The larger the area of those walls (A = l x h, in feet²) the higher the fee will be. However, both the number of walls and the area (l and h) have no effect on the price of the paint as there are not directly related. Increasing the area of the walls does not increase the quality (e.g. the price) of the paint. Likewise, painting more or less walls (i.e. varying n), does not suddenly change the quality or cost of the paint used. The dimension and quantity of the walls to be painted has not direct effect to the intrinsic value of the paint. Nonetheless, they can all be related through the equation:
Fee ($) = nKlh
We have now figured out that form the equation, the expression “nlh = nA” which gives us to the total area to be painted in a particular building, a quantity that does not speak of the type of paint brand used. This begs the question as to which factor then inform us of the quality of the paint or even the time it takes to complete the job both of which has the ability to increase or decrease the fee. The only factor left is K. In fact, K is measured as dollar per square foot. So the higher the K value, the more costly the painting service is per (total) area to be painted. Therefore, the painter is able to adjust the pricing for his service by modifying the value of K. The quality of the paint and other variable such as the time it will take for the painter to complete his takes can be factored into K, which is directly proportional to the fee. The answer is C).
