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Chapter 1: Problem 63

The stopping distance (at some fixed speed) of regular tires on glare ice is given by $$ D(F)=2 F+115 $$ where \(D(F)\) is the stopping distance, in feet, when the air temperature is \(F\), in degrees Fahrenheit. a) Find \(D\left(0^{\circ}\right), D\left(-20^{\circ}\right), D\left(10^{\circ}\right)\), and \(D\left(32^{\circ}\right)\). b) Explain why the domain should be restricted to the interval \(\left[-57.5^{\circ}, 32^{\circ}\right]\)

Short Answer

Expert verified
a) D(0) = 115, D(-20) = 75, D(10) = 135, D(32) = 179. b) Domain: \([-57.5^{\circ}F, 32^{\circ}F]\) to ensure non-negative distance.

Step by step solution

01

Evaluate D(F) at F = 0

Substitute 0 for F in the function D(F). D(0) = 2(0) + 115 = 115. So, D(0) = 115 feet.
02

Evaluate D(F) at F = -20

Substitute -20 for F in the function D(F). D(-20) = 2(-20) + 115 = -40 + 115 = 75. So, D(-20) = 75 feet.
03

Evaluate D(F) at F = 10

Substitute 10 for F in the function D(F). D(10) = 2(10) + 115 = 20 + 115 = 135. So, D(10) = 135 feet.
04

Evaluate D(F) at F = 32

Substitute 32 for F in the function D(F). D(32) = 2(32) + 115 = 64 + 115 = 179. So, D(32) = 179 feet.
05

Reason for Domain Restriction

D(F) must be a positive distance to make sense. The smallest value for \(D(F)\) should be zero: \(0 = 2F + 115\). Solving for F: \(0 = 2F + 115\). Therefore, \(F = -57.5\). Thus, the domain is restricted to \([-57.5^{\circ}F, 32^{\circ}F]\) to ensure stopping distance is non-negative.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

temperature conversion
Understanding temperature conversion is crucial for solving many real-world problems. There are multiple temperature scales like Fahrenheit and Celsius. Sometimes, we need to convert values between these scales. The formula to convert Celsius to Fahrenheit is given by: 掳F = (掳C * 9/5) + 32. To convert from Fahrenheit to Celsius, use the formula: 掳C = (掳F - 32) * 5/9. For example, if we want to convert 0掳C to Fahrenheit, we substitute it into the formula: 0掳C * 9/5 + 32 = 32掳F. This understanding can help us interpret the temperatures given in the formula for stopping distance carefully.
evaluating functions
Evaluating functions is a key step in solving mathematical problems. It translates abstract formulas into concrete numbers. In the given exercise, we have the function: D(F) = 2F + 115. To evaluate this function at specific temperatures, we simply substitute the values of F into the function.
For instance, when F = 0, the function evaluates as follows:
D(0) = 2(0) + 115 = 115 feet.
The same steps can be taken for other temperatures:
  • For F = -20: D(-20) = 2(-20) + 115 = 75 feet
  • For F = 10: D(10) = 2(10) + 115 = 135 feet
  • For F = 32: D(32) = 2(32) + 115 = 179 feet
Evaluating functions in this way provides the specific stopping distances for different temperatures.
domain restriction
Domain restriction is an important concept in ensuring that functions make sense in real-world contexts. The domain of a function is the set of all possible input values (in this case, temperatures) that produce valid outputs (non-negative stopping distances).
In the exercise, we restrict the domain to the values of F where D(F) is non-negative. We solve for F when D(F) = 0:
0 = 2F + 115
2F = -115
F = -57.5
Therefore, the domain is restricted to the interval [-57.5, 32] to ensure that stopping distances are non-negative. This interval represents the temperatures where the stopping distance remains practical and meaningful.
function interpretation
Function interpretation involves understanding what the output of a function means in its real-world context. In this exercise, D(F) represents the stopping distance of regular tires on glare ice as a function of air temperature in degrees Fahrenheit.
This tells us that as the temperature increases, the stopping distance also increases, due to the positive coefficient 2 in the function: D(F) = 2F + 115. This logical relationship helps us make predictions and informed decisions based on the temperatures:
  • At 0掳F, the stopping distance is 115 feet.
  • At -20掳F, the stopping distance is 75 feet.
  • At 10掳F, the stopping distance is 135 feet.
  • At 32掳F, the stopping distance is 179 feet.
Interpreting functions in this manner helps us understand and predict real-life phenomena effectively.

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