The area of an ellipse in the form (x*x/a*a)+(y*y)/(b*b)=1 is A=3.14*ab. For this ellipse, a+b=20.
a. write the area of the ellipse as a function of a.
b. Find the equation of the ellipse with an area of 264 square centimeters.
c. Complete the table and make a conjecture about the shape of the ellipse with a maximum area.
a 7 8 10 11 12 13 14
b
area
what is <mnp=60 degrees
a. To write the area of the ellipse as a function of a, you can use the given formula A = 3.14 * ab and the fact that a + b = 20.
Substitute the value of b = 20 - a, since a + b = 20, into the area formula:
A = 3.14 * a * (20 - a)
b. To find the equation of the ellipse with an area of 264 square centimeters, set A = 264 in the area formula:
264 = 3.14 * a * (20 - a)
Simplify and rearrange the equation:
264 = 62.8a - 3.14a^2
Rearrange the equation to standard form:
3.14a^2 - 62.8a + 264 = 0
This is a quadratic equation. You can solve it using factoring, completing the square, or the quadratic formula. Solving it, you will get two values for a. Since a and b are interchangeable for an ellipse, you will have two sets of a and b values that satisfy the equation.
c. Complete the table and make a conjecture about the shape of the ellipse with a maximum area.
To complete the table, substitute the given values of a into the equation A = 3.14 * a * (20 - a).
a 7 8 10 11 12 13 14
b
Using the equation a + b = 20, you can find the corresponding values of b for each given value of a.
For example, when a = 7, b = 20 - 7 = 13.
Complete the table with the calculated values of b:
a 7 8 10 11 12 13 14
b 13 12 10 9 8 7 6
Now, calculate the area for each pair of values (a, b) using the formula A = 3.14 * a * b:
a 7 8 10 11 12 13 14
b 13 12 10 9 8 7 6
area 286 301.44 314 297.66 288 279.22 268.08
From the table, we can observe that as a increases, the area first increases, reaches a maximum value, and then starts decreasing. Therefore, we can make a conjecture that the shape of the ellipse with a maximum area is symmetrical with respect to the x-axis.
(a) The area of an ellipse is
A = pi*a*b.
In this case,
A = pi * a * (20 - a)
(b) Set A = 264 and solve for a. Than use a+b=20. to get b, Then write the equation
(c) Complete the table using a+b=20. If a = 7, b = 13.
See which combination of a and b gives you a maximum value of a*b, and therefore of the area. I suspect it will be a = b = 10
This is work you should be able to do yourself.