For #11, the answers are choices 1 and 4. The first one is because it is a parallelogram so A = bh (12 x 3). The fourth choice is correct because it is a triangle with a base of 6 and a height of 12 and A = 1/2bh so 1/2 x 6 x 12 = 36.
For #12, you will find the areas of 2 rectangles.
1. A = 9 x 8
A = 20 x 9
72 + 180 = 252 square feet
For #13, you will use the formula for finding the area of a trapezoid and double it,
A = 1/2h(b1 + b2)
1/2 x 24 (40 + 60)
A = 120 square inches
120 x 2 = 240 square inches
For # 14, you will use the formula for finding the area of a trapezoid again.
A = 1/2 x 5 (9 + 11)
A = 50 square meters
50 x 2 = 100 square meters
For #15, the corner cabinet is the triangle in the corner.
Use the formula for finding the area of a triangle to find the space.
A = 1/2bh
1/2 x 3 x 3
A = 4.5 square feet
Answer:
45
Step-by-step explanation:
We are using sigma notation to solve for a sum of arithmetic sequences:
The 5 stands for stop at i = 5 (inclusive)
The i = 1 stands for start at i = 1
The 3i stands for expression of each term in the sum
15 wipe up in keep jdjdjdjd
The question is:
Consider the differential equation:
![y''-2y' + 17y = 0 \\ e^xcos 4x, e^x sin 4x\\ $on the interval$ (-\infty, \infty).](https://tex.z-dn.net/?f=y%27%27-2y%27%20%2B%2017y%20%3D%200%20%5C%5C%20e%5Excos%204x%2C%20e%5Ex%20sin%204x%5C%5C%20%24on%20the%20interval%24%20%28-%5Cinfty%2C%20%5Cinfty%29.)
(1) Verify that the given functions form a fundamental set of solutions of the differential equation on the indicated interval. The functions satisfy the differential equation and are linearly independent since ![W\left(e^x cos 4x, e^x sin 4x \right) \neq 0 \\ $on$ -\infty< x < \infty.](https://tex.z-dn.net/?f=W%5Cleft%28e%5Ex%20cos%204x%2C%20e%5Ex%20sin%204x%20%5Cright%29%20%5Cneq%200%20%5C%5C%20%24on%24%20-%5Cinfty%3C%20x%20%3C%20%5Cinfty.)
(2) Form the general solution.
Answer:
(1) To verify if the given functions form a fundamental set of solutions to the differential equation, we find the Wronskian of the two functions.
The Wronskian of functions
is given as
![W(y_1, y_2) = \left|\begin{array}{cc}y_1&y_2\\y_1'&y_2'\end{array}\right|\\\\y_1= e^x cos 4x \\y_2 = e^x sin 4x \\y_1' = -4e^x sin 4x \\y_2' = 4e^x cos 4x \\](https://tex.z-dn.net/?f=W%28y_1%2C%20y_2%29%20%3D%20%5Cleft%7C%5Cbegin%7Barray%7D%7Bcc%7Dy_1%26y_2%5C%5Cy_1%27%26y_2%27%5Cend%7Barray%7D%5Cright%7C%5C%5C%5C%5Cy_1%3D%20e%5Ex%20cos%204x%20%5C%5Cy_2%20%3D%20e%5Ex%20sin%204x%20%5C%5Cy_1%27%20%3D%20-4e%5Ex%20sin%204x%20%5C%5Cy_2%27%20%20%3D%204e%5Ex%20cos%204x%20%5C%5C)
![W\left(e^x cos 4x, e^x sin 4x \right) = \left|\begin{array}{cc}e^x cos 4x &e^x sin 4x \\ \\ -4e^x sin 4x&4e^x cos 4x \end{array}\right|](https://tex.z-dn.net/?f=W%5Cleft%28e%5Ex%20cos%204x%2C%20e%5Ex%20sin%204x%20%5Cright%29%20%3D%20%20%5Cleft%7C%5Cbegin%7Barray%7D%7Bcc%7De%5Ex%20cos%204x%20%26e%5Ex%20sin%204x%20%5C%5C%20%5C%5C%20-4e%5Ex%20sin%204x%264e%5Ex%20cos%204x%20%5Cend%7Barray%7D%5Cright%7C)
![= 4e^{2x} cos^2 4x + e^{2x} sin^2 4x \\ \\ = 4e^{2x}\left( cos^2 4x + sin^2 4x\right)\\ \\$but $cos^2 4x + sin^2 4x = 1\\ \\W\left(y_1, y_2 \right) = 4e^{2x} \neq 0](https://tex.z-dn.net/?f=%3D%20%204e%5E%7B2x%7D%20cos%5E2%204x%20%2B%20e%5E%7B2x%7D%20sin%5E2%204x%20%5C%5C%20%5C%5C%20%3D%204e%5E%7B2x%7D%5Cleft%28%20cos%5E2%204x%20%2B%20sin%5E2%204x%5Cright%29%5C%5C%20%5C%5C%24but%20%24cos%5E2%204x%20%2B%20sin%5E2%204x%20%3D%201%5C%5C%20%5C%5CW%5Cleft%28y_1%2C%20y_2%20%5Cright%29%20%3D%204e%5E%7B2x%7D%20%5Cneq%200)
(2) The general solution may be expressed as a linear combination
![y = C_1e^x cos 4x + C_2e^x sin 4x](https://tex.z-dn.net/?f=y%20%3D%20C_1e%5Ex%20cos%204x%20%2B%20C_2e%5Ex%20sin%204x)
Where
are arbitrary constants.