Question

1. math on the spot suppose the cost in dollars of producing x model cars is given by the polynomial 500,000 + 2x and the revenue generated from sales in dollars by the polynomial 30x - 0.0001x². find a polynomial expression for the profit from making and selling x model cars. then evaluate the expression for x = 300,000. 12. critique reasoning to solve the problem shown at the right, marla wrote the following solution. (4 - 3x + 2x²) + (x² - 7x + 5) explain the error in marlas reasoning. then correct the error and finish solving the problem. 13. suppose f(x) = 4x³ + 3x² - 12x - 32, h(x) = 4x⁴ - 3x² + 2x - 19, and (f + g)(x) = h(x). what is g(x)? a g(x) = -4x⁴ + 4x³ + 6x² - 14x - 13 b g(x) = 4x⁴ - 4x³ - 6x² + 14x + 13 c g(x) = 4x⁴ + 4x³ - 10x - 51 d g(x) = -4x⁴ - 4x² + 10x + 51 14. the sides of a triangle are represented by the expressions 5 - 4x², 32x - 5x², and x² - 5. what is the perimeter of the triangle? a -8x² + 32x b 2x² + 32x c -8x² + 32x + 10 d 2x² + 32x + 10

Explanation:

13.

Step1: Use the formula for function addition

Since \((f + g)(x)=h(x)\), then \(g(x)=h(x)-f(x)\).

Step2: Substitute the given functions

\(h(x)=4x^{4}-3x^{2}+2x - 19\) and \(f(x)=4x^{3}+3x^{2}-12x - 32\). So \(g(x)=(4x^{4}-3x^{2}+2x - 19)-(4x^{3}+3x^{2}-12x - 32)\).

Step3: Distribute the negative sign

\(g(x)=4x^{4}-3x^{2}+2x - 19-4x^{3}-3x^{2}+12x + 32\).

Step4: Combine like - terms

For the \(x^{4}\) term: \(4x^{4}\).
For the \(x^{3}\) term: \(-4x^{3}\).
For the \(x^{2}\) term: \(-3x^{2}-3x^{2}=-6x^{2}\).
For the \(x\) term: \(2x + 12x=14x\).
For the constant term: \(-19 + 32 = 13\).
So \(g(x)=4x^{4}-4x^{3}-6x^{2}+14x + 13\).

Step1: Recall the formula for the perimeter of a triangle

The perimeter \(P\) of a triangle with side lengths \(a\), \(b\), and \(c\) is \(P=a + b + c\). Here \(a = 5-4x^{2}\), \(b=32x-5x^{2}\), and \(c=x^{2}-5\).

Step2: Add the three expressions

\(P=(5-4x^{2})+(32x-5x^{2})+(x^{2}-5)\).

Step3: Combine like - terms

For the \(x^{2}\) terms: \(-4x^{2}-5x^{2}+x^{2}=-8x^{2}\).
For the \(x\) terms: \(32x\).
For the constant terms: \(5-5 = 0\).

Answer:

B. \(g(x)=4x^{4}-4x^{3}-6x^{2}+14x + 13\)

14.