A basketball team plays in a stadium that holds 60,000 people. With ti

To maximize the product of two numbers, they ought to be equal.
Bunuel would approve.

To maximize the product xy, both the terms must be brought close to each other

For every dollar reduction in price, attendance increases by 1k

(30-x)(30+x)*10^3

30-x=30+x = 30


Answer C

If quick with calculations, one can find the “slope” or constant of proportionality and then test each answer choice, finding the highest value.

The concept behind the problem:

The relationship between ticket prices and attendance is a linear relationship.

This means, the relationship between the X variable (ticket price) and the Y variable (attendance) can be expressed with a Linear equation (ie, a function that represents a straight line in the coordinate plane)

(1st) slope

In order to simplify calculation, turn the attendance figure into 1,000 Units

36,000 people = 36 “thousand units
32,000 people = 32 “thousand units”

($24 , 36)

($28 , 32)

As the ticket price rises by + $4 (along the X-axis)

The attendance DROPS by — 4 k (along the Y-axis)

Slope = -4 / 4 = -1

y = (-1)(x) + b

Where b = y intercept

Plug in a coordinate: ($24 , 36)

36 = (-1)(24) + b

b = 36 + 24 = 60

Linear Equation that governs the linear relationship is:

y = -x + 60

or

y + x = 60

Where:

X = the ticket price offered
And
Y = the number of people in attendance who pay the price of that ticket, in “1,000 units”

(2nd) concept:

Given a constant addition of two variables that must be +positive real numbers, the MAXIMUM product is obtained when the two positive variables are equal

Therefore, GIVEN

X + Y = 60

The MAXIMUM Product will be obtained ($ticket price * attendance) when:

X = Y = 30

$30 * (30 k) = $30 * (30, 000) = $900,000 is the maximum revenue that can be obtained


Answer C— $30

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A few posts above have mentioned the "rule," but I thought it might be useful to cover how we know that 30*30 is going to be more than 28*32. Let's give ourselves easier numbers so we can see it work.

Look at the following:
10*0 = 0
9*1 = 9
8*2 = 16
7*3 = 21
6*4 = 24
5*5 = 25

If we are looking to MAXIMIZE, we want to pull the two numbers closer to being the same value. If we are looking to MINIMIZE, we want to push them farther apart. I think the most common way GMAC has tested this concept over the years is with questions that ask us to either maximize or minimize the area of a rectangle (common phrasing is the size of a plot of land or a room) within some constraints, typically that the perimeter is defined and we need to decide whether we want the longest/skinniest rectangle allowed or the most-squarelike rectangle allowed.

The revenue question here is a good version of the concept. 28*32(k) is good. Can we do better? Yes, 30*30(k).

Answer choice C.

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