Chapter 4.4, Problem 80E

### Calculus: Early Transcendental Fun...

7th Edition
Ron Larson + 1 other
ISBN: 9781337552516

Chapter
Section

### Calculus: Early Transcendental Fun...

7th Edition
Ron Larson + 1 other
ISBN: 9781337552516
Textbook Problem

# Specific Gravity A model for the specific gravity of water S is S = 5.755 10 8 T 3 − 8.521 10 6 T 2 + 6.540 10 5 T + 0.99987 ,     0 < T < 25 where T is the water temperature in degrees Celsius.(a) Use the second derivative to determine the concavity of S.(b) Use a computer algebra system to find the coordinates of the maximum value of the function.(c) Use a graphing utility to graph the function over the specified domain. (Use a setting in which 0.996 ≤ S ≤ 1.001 .)(d) Estimate the specific gravity of water when T = 20 ∘ .

(a)

To determine

To calculate: The concavity of the provided function

S=5.755108t38.521106t2+6.540105t+0.99987.

Explanation

Given: The function that gives the specific gravity of water at t degree Celsius.

S=5.755108t3âˆ’8.521106t2+6.540105t+0.99987

Here t is between 0 and 25.

Formula used: The test for concavity states that, for a function f that is twice differentiable on an open interval I:

If f''(x)>0 for all x in I, the function f is concave upwards on I and if f''(x)<0 for all x in I, the function f is concave downwards on I.

Calculation: Differentiate the function twice to obtain:

S'(t)=3(5.755)108t2âˆ’2(8.521)106t+6.540105S''(t)=6(5

(b)

To determine

To calculate: The coordinates of the maximum value of the function

S=5.755108t38.521106t2+6.540105t+0.99987.

(c)

To determine

To graph: The function S=5.755108t38.521106t2+6.540105t+0.99987.

(d)

To determine

To calculate: The approximate specific gravity when the temperature is 20 degrees Celsius.

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