The table shows the approximate height of an object [tex]$x$[/tex] seconds after the object was dropped. The function [tex]f(x) = -16x^2 + 100$[/tex] models the data in the table.
Height of a Dropped Object
| Time (seconds) | Height (feet) |
| -------------- | ------------- |
| 0 | 100 |
| 0.5 | 96 |
| 1 | 84 |
| 1.5 | 65 |
| 2 | 37 |
For which value of [tex]$x$[/tex] would this model make the least sense to use?
A. -2.75
B. 0.25
C. 1.75
D. 2.25
Answer (2)
The model makes the least sense when the height is negative, so we need to find when − 16 x 2 + 100 < 0 .
Solving the inequality, we get \frac{25}{4}"> x 2 > 4 25 , which means 2.5"> x > 2.5 or x < − 2.5 .
Checking the given values, only x = − 2.75 satisfies this condition.
Therefore, the model makes the least sense for − 2.75 .
Explanation
Understanding the Problem We are given a function f ( x ) = − 16 x 2 + 100 that models the height of a dropped object x seconds after it was dropped. We want to find the value of x for which this model makes the least sense to use. Since the height of an object cannot be negative, the model makes the least sense when the height is negative.
Finding the Critical Values We need to find the value of x for which f ( x ) = − 16 x 2 + 100 < 0 . This means − 16 x 2 < − 100 , or 100"> 16 x 2 > 100 . Dividing both sides by 16, we get \frac{100}{16} = \frac{25}{4}"> x 2 > 16 100 = 4 25 . Taking the square root of both sides, we have \frac{5}{2} = 2.5"> ∣ x ∣ > 2 5 = 2.5 . This means 2.5"> x > 2.5 or x < − 2.5 .
Checking the Given Values Now we check which of the given values satisfy 2.5"> x > 2.5 or x < − 2.5 . The given values are − 2.75 , 0.25 , 1.75 , and 2.25 .
For x = − 2.75 , we have x < − 2.5 , so this value satisfies the condition.
For x = 0.25 , we have x ≮ − 2.5 and x ≯ 2.5 , so this value does not satisfy the condition.
For x = 1.75 , we have x ≮ − 2.5 and x ≯ 2.5 , so this value does not satisfy the condition.
For x = 2.25 , we have x ≮ − 2.5 and x ≯ 2.5 , so this value does not satisfy the condition.
Therefore, the only value that satisfies the condition is x = − 2.75 .
Evaluating the Function Since the model makes the least sense when the height is negative, we can evaluate the function at each of the given values:
f ( − 2.75 ) = − 16 ( − 2.75 ) 2 + 100 = − 16 ( 7.5625 ) + 100 = − 121 + 100 = − 21
f ( 0.25 ) = − 16 ( 0.25 ) 2 + 100 = − 16 ( 0.0625 ) + 100 = − 1 + 100 = 99
f ( 1.75 ) = − 16 ( 1.75 ) 2 + 100 = − 16 ( 3.0625 ) + 100 = − 49 + 100 = 51
f ( 2.25 ) = − 16 ( 2.25 ) 2 + 100 = − 16 ( 5.0625 ) + 100 = − 81 + 100 = 19
The value of x for which the model makes the least sense is x = − 2.75 , since the height is negative at this time.
Final Answer The model makes the least sense for x = − 2.75 .
Examples
Understanding how mathematical models apply to real-world scenarios is crucial in many fields. For instance, in physics, models like the one in this problem help predict the motion of objects. However, these models have limitations. For example, the height of the object cannot be negative, and time cannot be negative in this context. Recognizing when a model's assumptions are violated, such as predicting a negative height or time, is essential for making accurate predictions and informed decisions. This is also applicable in finance when predicting stock prices or in engineering when designing structures.
The model is least applicable for x = − 2.75 because that leads to a negative height, which is physically meaningless. For other values, the height remains valid. Thus, the answer is − 2.75 .
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