Chapter 5.1, Problem 105E

### Calculus (MindTap Course List)

11th Edition
Ron Larson + 1 other
ISBN: 9781337275347

Chapter
Section

### Calculus (MindTap Course List)

11th Edition
Ron Larson + 1 other
ISBN: 9781337275347
Textbook Problem

# Modeling Data The table shows the temperatures T (in degrees Fahrenheit) at which water boils at selected pressures p (in pounds per square inch). (Source: Standard Handbook of Mechanical Engineers) p 5 10 14.696(1 atm) 20 T 162.24 193.21 212.00 227.96 p 30 40 60 80 100 T 250.33 267.25 292.71 312.03 327.81 A model that approximates the data is T = 87.97 + 34.96 ln p + 7.91 p (a) Use a graphing utility to plot the data and graph the model.(b) Find the rates of change of T with respect to p when p = 10 and p = 70 .(c) Use a graphing utility to graph T'. Find lim p → ∞ T ' ( p ) and interpret the result in the context of the problem.

(a)

To determine

To graph: The model, T=87.97+34.96lnp+7.91p and the table shows the temperatures T (in degrees

Fahrenheit) at which water boils at selected pressures p (in pounds per square inch).

 p 5 10 14.696 (1 atm) 20 T 162.24 193.21 212.00 227.96
 p 30 40 60 80 100 T 250.33 267.25 292.71 312.03 327.81

The model approximates the data as T=87.97+34.96lnp+7.91p.

Explanation

Given:

The model, the table shows the temperatures T (in degrees Fahrenheit) at which water boils at selected pressures p (in pounds per square inch).

 p 5 10 14.696 (1 atm) 20 T 162.24 193.21 212.00 227.96
 p 30 40 60 80 100 T 250.33 267.25 292.71 312.03 327.81

The model approximates the data as T=87.97+34.96lnp+7.91p.

Graph:

Use the graphing calculator to graph the model, the table shows the temperatures T (in degrees Fahrenheit) at which water boils at selected pressures p (in pounds per square inch).

 p 5 10 14.696 (1 atm) 20 T 162.24 193.21 212.00 227.96
 p 30 40 60 80 100 T 250.33 267.25 292.71 312.03 327

(b)

To determine

To calculate: The rate of change of T with respect to p when p=10 and p=70 in the model, the table shows the temperatures T (in degrees Fahrenheit) at which water boils at selected pressures p (in pounds per square inch)

 p 5 10 14.696 (1 atm) 20 T 162.24 193.21 212.00 227.96
 p 30 40 60 80 100 T 250.33 267.25 292.71 312.03 327.81

The model approximates the data as T=87.97+34.96lnp+7.91p.

(c)

To determine

To graph: The derivative T=34.96p+3.955p evaluated in part (b) in the model T=87.97+34.96lnp+7.91p and, the table shows the temperatures T (in degrees Fahrenheit) at which water boils at selected pressures p (in pounds per square inch).

 p 5 10 14.696 (1 atm) 20 T 162.24 193.21 212.00 227.96
 p 30 40 60 80 100 T 250.33 267.25 292.71 312.03 327.81

The model approximates the data as T=87.97+34.96lnp+7.91p.

Also, find limpT(p).

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