303,718 views
30 votes
30 votes
A carnival game involves striking a lever that forces a weight up a tube. If the weight reaches 20 feet to ring the bell, the contestant wins a prize. The function h(t) = -16t2 + 32t + 3 gives the height of the weight at any given time. Do not round the answer.Find the maximum height of the weight. How many seconds will it take for the weight to reach the maximum height? Will the contestant win a prize?

User Nithin Satheesan
by
2.7k points

1 Answer

26 votes
26 votes

To answer this question, we can see that the function that gives the height of the weight at any given time is a parabola with a maximum (the negative number in front of the leading term gives us this information):


h(t)=-16t^2+32t+3

We can find the maximum height of the weight by finding the vertex of the parabola - we already know that this vertex is a maximum in this case.

The formula to find the vertex of a parabola is given by:


\begin{gathered} x_v=-(b)/(2a) \\ y_v=c-(b^2)/(4a) \end{gathered}

This is for the next general equation:


ax^2+bx+c

Now, we have that the parabola is:


h(t)=-16t^2+32t+3

Then we have:


\begin{gathered} a=-16 \\ b=32 \\ c=3 \end{gathered}
\begin{gathered} x_v=-(b)/(2a) \\ x_v=-(32)/(2(-16))=-(32)/(-32)=1 \\ x_v=1 \end{gathered}

And


\begin{gathered} y_v=c-(b^2)/(4a) \\ y_v=3-((32)^2)/(4(-16)) \\ y_v=3-(32^2)/(-64) \\ y_v=3+(32\cdot32)/(2\cdot32)=3+(32)/(2)\cdot(32)/(32)\Rightarrow(32)/(32)=1,(a)/(a)=1 \\ y_v=3+(32)/(2) \\ y_v=3+16=19 \\ y_v=19 \end{gathered}

Therefore, we have that the vertex of the parabola is (1, 19). That is, we have that the maximum point for this parabola is y = 19 - this is the maximum value for the function, and the function takes this value when x = 1.

In other words, in summary, we have:

• The maximum height of the weight is 19 feet (we assume the function gives us the height in feet.)

,

• The weight will take 1 second to reach the maximum height (we assume that the function is expressed in seconds.)

,

• The contestant will not win a prize since the weight will never reach 20 feet.

User Vincent Tjeng
by
2.7k points