Anonymous
Anonymous asked in Science & MathematicsMathematics · 1 month ago

# confusing math practice sat question? Relevance

The function P(x) = -2x² + 20x - 48 represents a parabola.

Since the leading coefficient (-2) is negative, we know we have a downward facing parabola, so the vertex will be a maximum.

In general, the x-coordinate of the vertex of a parabola (ax² + bx + c) can be found using:

x = -b/(2a)

x = -20/(2 * -2)

x = -20/-4

x = 5

Plugging in x = 5, we get:

P(5) = -2(5²) + 20(5) - 48

P(5) = -50 + 100 - 48

P(5) = 2

That's the company's profit in *thousands* of dollars.

Another way to get that result is to factor the quadratic:

P(X) = -2(x² - 10x + 24)

P(X) = -2(x - 4)(x - 6)

The two roots are x = 4 and x = 6. We know the line of symmetry is halfway between these two roots, so we can mentally figure out it's 5. (Alternatively, take the average of 4 and 6 --> (4+6)/2 = 5)

Then calculate P(5) as before.

Another method, if you know calculus is to take the derivative.

P'(x) = -4x + 20

Set it to zero:

-4x + 20 = 0

-4x = -20

x = -20/-4

x = 5

However you get to x=5, calculate P(5) = 2 and multiply it by 1000.

• P(x) = -2x² + 20x - 48

or, P(x) = -2(x² - 10x) - 48

so, P(x) = -2(x - 5)² - 48 + 50

Hence, P(x) = -2(x - 5)² + 2

We can see that x value of x = 5 gives P(5) = 2

Therefore, selling an item for \$5.00 will yield a maximum profit of \$2000.

:)>

• P(x) = -2x^2 +20x -48.;

P'(x) = -4x+20, = 0 at [x,P(x)] = (5,2).;

P''(x) = -4, < 0----> any extremum is a maximum.;

Clearly, max profit occurs where P(x) = 2 which corresponds to choice B)

• You don't need calculus for this one.  The polynomial factors pretty easily:

P(x) = -2 (x² - 10x + 24)         . . . . common factor of -2

P(x) = -2 (x - 4) (x - 6)            . . . . leaving an easy quadratic to factor

The graph of that is a parablola, concave downward because of the negative leading coefficient (-2), with a maximum halfway between the roots at x=4 and x=6.

The maximum value of P(x) occurs at x=5 (halfway between 4 and 6) and the profit at that point is:

P(5) = -2(5²) + 20(5) - 48 = -50 + 100 - 48 = 2 thousand dollars.

If you don't care for graphical reasoning, you could complete the square instead:

P(x)  = -2(x² - 10x + 24)

=  -2(x² - 10x + 25 - 25 + 24)   .... add and subtract (-10/2)² = 25

= -2[(x - 5)² - 25 + 24]              .... factor the first 3 terms as a perfect square

= -2(x - 5)² + 2                         .... and simplify

The -2(x - 5)² term is either negative or zero, with maximum of zero, and that happens when x-5 = 0, or x=5.  And the + 2 constant term is the (maximum) profit when the squared term is zero.  That's 2 thousand dollars, just as above.