Please help me with the following 2 questions.
1. Find the area of the curved surface of a right-circular cone of radius 6 and height 4 by rotating the straight line segment from (0,0) to (6,4) about the y-axis.
2. Use the Maclaurin polynomial of degree 4 to approximate sin(0.2). The formula given is the sum between k = 0 and infinity of [-1^k]/(2k+1)!x^(2k+1). I tried using the following but got it wrong
0.2 - (0.2)^3/3! + (0.2)^5/5! - (0.2)^7/7! + (0.2)^9/9!.
Please help.
Thanks
The area of a cone of radius r and slant height s is πrs
r = 6
s^2 = 6^2 + 4^2
s = √52 = 2√13
area = 12π√13
For MacLaurin polynomials, I may be wrong, but I thought you wanted degree 4. That would just be
0 + 1/1! x + 0/2! x^2 + -1/3! x^3 + 0/4! x^4
= x - x^3/3!
= .2 - .008/6 = 0.1987
Thanks Steve.
Sure! I'll help you with both questions.
1. To find the area of the curved surface of a right-circular cone, you can use the formula A = πrs, where r is the radius and s is the slant height. In this case, the slant height can be found using the Pythagorean theorem: s = √(r^2 + h^2), where h is the height of the cone.
So, let's calculate the slant height:
h = 4 (given height of the cone)
r = 6 (given radius of the cone)
s = √(6^2 + 4^2) = √(36 + 16) = √(52) = 2√13
Now, we can calculate the area of the curved surface:
A = π * r * s
= π * 6 * 2√13
= 12π√13
Therefore, the area of the curved surface of the cone is 12π√13.
2. The Maclaurin polynomial of degree 4 for sin(x) is given by the series:
sin(x) = x - (x^3/3!) + (x^5/5!) - (x^7/7!) + ...
To approximate sin(0.2) using this polynomial, we can use the terms up to the fourth degree.
sin(0.2) ≈ 0.2 - (0.2^3/3!) + (0.2^5/5!) - (0.2^7/7!)
Now let's calculate each term:
0.2^3 = 0.008
3! = 3 * 2 * 1 = 6
0.2^5 = 0.00032
5! = 5 * 4 * 3 * 2 * 1 = 120
0.2^7 = 0.000008
7! = 7 * 6 * 5 * 4 * 3 * 2 * 1 = 5040
Using these values, we can plug them into the formula:
sin(0.2) ≈ 0.2 - (0.008/6) + (0.00032/120) - (0.000008/5040)
Now, perform the calculations:
sin(0.2) ≈ 0.2 - 0.001333 + 0.000002667 - 0.0000000015873
Finally, add all the terms together:
sin(0.2) ≈ 0.1986682673
Therefore, the approximation of sin(0.2) using the Maclaurin polynomial of degree 4 is approximately 0.1987.