Question

the height, h, of a ball is given by ( h(t) = -2t^2 - 12t - 10 )
a. give the polynomial in factored form. ( h(t) = )

b. determine the times, t, when the ball hits the ground. (type the smaller t - value in the first blank, then the larger t - value in the second blank.)
( t = ) ( t = )

c. do both the t - values make sense in the context of the problem? (type y or n)

pregunta 26 4 puntos
a company’s profit is modeled by ( p(x) = -x^2 + 10x - 16 ), where x is the number of items sold in hundreds.
a. factor the quadratic. ( p(x) = )

b. what are the x - values that find the number of items sold that result in zero profit. (type the smaller x - value in the first blank, then the larger x - value in the second blank.)
( x = ) ( x = )

c. true or false. the company breaks even (profit = 0) if it sells 200 items or 800 items. (type t or f)

pregunta 27 1 puntos
what is the difference between a zero and an x - intercept of a polynomial function?

• a zero is a number that makes the polynomial zero; an x - intercept is a point on the graph.
• a zero is always positive; an x - intercept can be negative.
• a zero is the slope of the graph; an x - intercept is the maximum point.
• there is no difference; zeros and x - intercepts refer to the same values.

pregunta 28 2 puntos
which quadratic equation has 5 and - 4 as their solutions?

• ( x^2 + x + 20 = 0 )
• ( x^2 + 9x - 20 = 0 )
• ( x^2 - x - 20 = 0 )
• ( x^2 - 9x - 20 = 0 )

Explanation:

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First Problem (Ball Height)

Step1: Factor out common term

$h(t) = -2(t^2 + 6t + 5)$

Step2: Factor quadratic inside parentheses

$t^2 + 6t +5=(t+1)(t+5)$, so $h(t)=-2(t+1)(t+5)$

Step3: Set height to 0 for ground hit

$0=-2(t+1)(t+5)$

Step4: Solve for t-values

$t+1=0 \implies t=-1$; $t+5=0 \implies t=-5$

Step5: Check context validity

Time cannot be negative, so both values are invalid.

Step1: Factor out negative sign

$P(x) = -(x^2 -10x +16)$

Step2: Factor quadratic inside parentheses

$x^2 -10x +16=(x-2)(x-8)$, so $P(x)=-(x-2)(x-8)$

Step3: Set profit to 0 for break-even

$0=-(x-2)(x-8)$

Step4: Solve for x-values

$x-2=0 \implies x=2$; $x-8=0 \implies x=8$

Step5: Verify item count

$x=2$ means 200 items, $x=8$ means 800 items.

A zero is a numerical input that makes the polynomial equal to 0. An x-intercept is the coordinate point $(x,0)$ on the graph of the function where the output is 0.

Answer:

A. $-2(t+1)(t+5)$
B. $t=-5$, $t=-1$
C. N

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Pregunta 26 (Company Profit)