Concept explainers
Maximizing Area A wire 10 cm long is cut into two pieces, one of length x and the other of length 10 − x, as shown in the figure. Each piece is bent into the shape of a square.
- (a) Find a function that models the total area enclosed by the two squares.
- (b) Find the value of x that minimizes the total area of the two squares.
(a)
To find: The function that models the total area enclosed by two squares.
Answer to Problem 24P
The function that models the area of both the squares is
Explanation of Solution
Given:
Length of wire is
Length of first wire after cutting is x and length of second wire is
Calculation:
If length of wires are
Total enclosed area
Where,
Area of a square is,
Where, l is side of square.
Substitute
The area enclosed by square of side
Substitute
The total area is calculated as follows,
Further solving,
Thus, the function that models the area of both the squares is
(b)
To find: The value of
Answer to Problem 24P
The value of x that minimizes the area of the square is 5 cm.
Explanation of Solution
Function that models the area of square as calculated in part (a) is
Sketch the function
From the above figure, it can be observed that the function has minimum value at
Therefore, for the area to be minimum the length of the first wire must be 5 cm.
Chapter 2 Solutions
Precalculus: Mathematics for Calculus - 6th Edition
- Calculus: Early TranscendentalsCalculusISBN:9781285741550Author:James StewartPublisher:Cengage LearningThomas' Calculus (14th Edition)CalculusISBN:9780134438986Author:Joel R. Hass, Christopher E. Heil, Maurice D. WeirPublisher:PEARSONCalculus: Early Transcendentals (3rd Edition)CalculusISBN:9780134763644Author:William L. Briggs, Lyle Cochran, Bernard Gillett, Eric SchulzPublisher:PEARSON
- Calculus: Early TranscendentalsCalculusISBN:9781319050740Author:Jon Rogawski, Colin Adams, Robert FranzosaPublisher:W. H. FreemanCalculus: Early Transcendental FunctionsCalculusISBN:9781337552516Author:Ron Larson, Bruce H. EdwardsPublisher:Cengage Learning