a.
To show: that the area of the triangle is:
a.
Explanation of Solution
Given information: Let P(x, a) and Q (-x, a ) be two points on the upper half of the ellipse.
Centered at (0, 5). A triangle RST is formed by using the tangent lines to the ellipse at Q and P.
Calculation:
Let
The height h of the triangle is the y- intercept of the tangent line so substitute in x =0 into the tangent line equation to find the y-intercept. The height of the triangle is then;
Hence proved.
b.
To find: the domain of A and draw the graph of A and how are the asymptotes of the graph related to the problem situation.
b.
Answer to Problem 65E
Asymptotes: x
Explanation of Solution
Given information: Let P(x, a) and Q (-x, a ) be two points on the upper half of the ellipse.
Centered at (0, 5). A triangle RST is formed by using the tangent lines to the ellipse at Q and P.
Calculation:
To graph, plug in
The domain of A(x) is the intersection of the domains of the f(x) and f’(x) for which
There are vertical asymptotes x
c.
To find: the height of the triangle with minimum area and how is it related to the y-coordinates of the center of the ellipse.
c.
Answer to Problem 65E
Height of 15, its 3 times the y-coordinate of the center.
Explanation of Solution
Calculation:
Plugged in
To find the height , use
First find the values of f(x) and f’(x) at x =8.66 using calculator.
The height of the triangle with minimum area is then 15. Since the y- coordinate of the center of the ellipse is 5, this is 3 times the y- coordinates.
Therefore, height of 15 , its 3 times the y-coordinate of the center.
d.
To repeat: parts (a) − (c ) for the ellipse
Centered (0, B) and show that the triangle has minimum area when its height is 3B.
d.
Answer to Problem 65E
Height of the triangle is 3b which is three times the y − coordinate of the center of the ellipse.
Explanation of Solution
Calculation:
The height of the triangle can be found from the equation of the line tangent to point
P. y = height.
Therefore, height of the triangle is 3b which is three times the y − coordinate of the center of the ellipse.
Chapter 5 Solutions
Calculus: Graphical, Numerical, Algebraic: Solutions Manual
Additional Math Textbook Solutions
University Calculus: Early Transcendentals (4th Edition)
Calculus & Its Applications (14th Edition)
Precalculus
Precalculus: Concepts Through Functions, A Unit Circle Approach to Trigonometry (4th Edition)
University Calculus: Early Transcendentals (3rd 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