Question

Can you help me with 3 Algebra 2 questions?!?

1. Dalco Manufacturing estimates that its weekly profit, P, in hundreds of dollars, can be approximated by the formula p=-3x^2+6x+10, where x is the number of units produced per week, in thousands.

a. How man units should the company produce per week to earn the maximum profit?

b. Find the maximum weekly profit.

Choices:

1,000 units; $1,300

3,000 units; $100

1,000 units; $600

2,000 units; $1,100

2. Write the equation of the parabola in vertex form.

vertex (0,3), point (-4,-45)

Choices:

y=3(x-2)^2 + 2

y=3(x-2)^2 - 2

y=3(x+2)^2 + 2

y=(x+2)^2 + 2

3. Solve the equation by finding square roots.

108x^2 = 147

No choices!

Thanks and can anyone tell me how to factor using the X and the Box? I learned it 2 years ago and I have not used it for a while so if anyone knows can they use and example?! Thanks!

Answers

1. P = -3x2 + 6x + 10

a. With calculus, you'd take the deriviative, set it equal to zero, and solve for "x" to find the maximum profit. With algebra, we'll find the two roots (x-intercepts); the x-value of the maximum profit will be the average of the two roots:

-3x2 + 6x + 10 = 0

Complete the square to find the roots:

-3x2 + 6x = -10

(-3/-3)x2 + (6/-3)x = -10/-3

x2 - 2x = 10/3

x2 - 2x + (2/2)2 = 10/3 + (2/2)2

(x - 1)2 = 10/3 + 3/3 = 13/3

√(x - 1)2 = ±√(13/3)

x - 1 = ±√(13/3)

x = 1 ± √(13/3)

The two roots are: x = 1 - √(13/3), and x = 1 + √(13/3)

The average is:

x = [1 - √(13/3) + 1 + √(13/3)]/2 = 1

Therefore, the maximum profit occurs when x = 1 (1,000 units).

b. The maximum profit, in hundreds of dollars, is:

P = -3x2 + 6x + 10 = -3(1)2 + 6(1) + 10 = 13 = $1,300

The first choice is the correct answer.

2. A parabola in vertex form is y = a(x - h)2 + k, where "a" is a constant, and "h" and "k" are from the vertex (h,k). Given a vertex of (0,3), the equation becomes:

y = a(x - 0)2 + 3 = ax2 + 3

Substitute the x- and y-values from pont (-4,-45) into the equation and solve for the constant, a:

-45 = a(-4)2 + 3

16a = -48

a = -48/16 = -3

Thus, the equation for the parabola in vertex form is:

y = -3x2 + 3

3. 108x2 = 147

x2 = 147/108 = 49/36

x = ±√(49/36) = ±7/6

Thus, the two solutions are: x = -7/6, and x = 7/6.

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