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Subject Test Math 1

Question 29 of 32

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SAT Subject Test

The function h given by h left parenthesis t right parenthesis equals negative 16 t to the power of 2 end exponent plus 46 t plus 5 represents the height of a ball, in feet, t seconds after it is thrown. To the nearest foot, what is the maximum height the ball reaches?

Select an Answer

5

Correct Answer: 
No

23

Correct Answer: 
No

35

Correct Answer: 
No

38

Correct Answer: 
Yes

46

Correct Answer: 
No

The correct answer is D.

One way to determine the maximum height the ball reaches is to rewrite the quadratic expression that defines the function h by completing the square:

table row cell negative 16 t squared plus 46 t plus 5 end cell equals cell negative 16 open parentheses t squared minus 23 over 8 t close parentheses plus 5 end cell end table
space space space space space space space space space space space space space space space space space space space space space space space space space table row equals cell negative 16 open parentheses t squared minus 23 over 8 t plus open parentheses 23 over 16 close parentheses squared minus open parentheses 23 over 16 close parentheses squared close parentheses plus 5 end cell end table
table row cell space space space space space space space space space space space space space space space space space space space space space space space end cell equals cell negative 16 end cell end table open parentheses open parentheses t minus 23 over 16 close parentheses squared minus open parentheses 23 over 16 close parentheses squared close parentheses plus 5
table row cell space space space space space space space space space space space space space space space space space space space space space space space end cell equals cell negative 16 open parentheses t minus 23 over 16 close parentheses squared plus 16 open parentheses 23 over 16 close parentheses squared plus 5 end cell end table


It is not necessary to simplify any further, as the maximum height must correspond to t equals fraction numerator 23 over denominator 16 end fraction, which is the only value of t that makes the term negative 16 left parenthesis t minus fraction numerator 23 over denominator 16 end fraction right parenthesis to the power of 2 end exponent nonnegative. By substitution, h left parenthesis fraction numerator 23 over denominator 16 end fraction right parenthesis equals 16 left parenthesis fraction numerator 23 over denominator 16 end fraction right parenthesis to the power of 2 end exponent plus 5 equals 38.0625. Therefore, to the nearest foot, the maximum height the ball reaches is 38 feet.

 

Alternatively, one can use a graphing calculator to determine the maximum value of the function h. Since h left parenthesis 0 right parenthesis equals 5 comma h left parenthesis 1 right parenthesis equals negative 16 plus 46 plus 5 equals 35, h left parenthesis 2 right parenthesis equals negative 16 left parenthesis 4 right parenthesis plus 46 left parenthesis 2 right parenthesis plus 5 equals 33 and h left parenthesis 3 right parenthesis equals negative 16 left parenthesis 9 right parenthesis plus 46 left parenthesis 3 right parenthesis plus 5 equals negative 1, the maximum value of h must occur for some t minus text value end text between 1 and 2. Set the window so that the independent variable goes from 0 to 3 and the dependent variable goes from 0 to 50 to view the vertex of the parabola. Upon tracing the graph, the maximum value of h is slightly greater than 38. Therefore, to the nearest foot, the maximum height the ball reaches is 38 feet.

Question Difficulty: 
hard