Answer:
![y-2 = -3(x-2)](https://tex.z-dn.net/?f=y-2%20%3D%20-3%28x-2%29)
Step-by-step explanation:
Find the diagram to the question attached.
The standard equation of a line in point-slope form is expressed as:
where:
m is the slope of the line
is coordinate point on the line
First is to get the slope of the line.
![m = \frac{y_2-y_1}{x_2-x_1}](https://tex.z-dn.net/?f=m%20%3D%20%5Cfrac%7By_2-y_1%7D%7Bx_2-x_1%7D)
If after buying 2 lattes, she has $2 left, hence one of the coordinate of the line is (2, 2). We can get the other coordinates from the graph. From the graph, it can be seen that when x = 0, y = 8. Hence the other coordinates is (0, 8)
![m = \frac{8-2}{0-2}\\m = \frac{6}{-2}\\m = -3](https://tex.z-dn.net/?f=m%20%3D%20%5Cfrac%7B8-2%7D%7B0-2%7D%5C%5Cm%20%3D%20%5Cfrac%7B6%7D%7B-2%7D%5C%5Cm%20%3D%20-3)
Hence the slope is -3
Substitute m = -3 and the point (2, 2) into the formula to get the required equation (note that any of the points can be used)
![y-y_0 = m(x-x_0)\\y-2 = -3(x-2)](https://tex.z-dn.net/?f=y-y_0%20%3D%20m%28x-x_0%29%5C%5Cy-2%20%3D%20-3%28x-2%29)
Hence the equation of the line in point-slope form is ![y-2 = -3(x-2)](https://tex.z-dn.net/?f=y-2%20%3D%20-3%28x-2%29)
Answer:
![\displaystyle \int\limits^2_0 {f(x)} \, dx = \frac{159}{10}](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cint%5Climits%5E2_0%20%7Bf%28x%29%7D%20%5C%2C%20dx%20%3D%20%5Cfrac%7B159%7D%7B10%7D)
General Formulas and Concepts:
<u>Calculus</u>
Integration
- Integrals
- Integral Notation
Integration Rule [Reverse Power Rule]: ![\displaystyle \int {x^n} \, dx = \frac{x^{n + 1}}{n + 1} + C](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cint%20%7Bx%5En%7D%20%5C%2C%20dx%20%3D%20%5Cfrac%7Bx%5E%7Bn%20%2B%201%7D%7D%7Bn%20%2B%201%7D%20%2B%20C)
Integration Rule [Fundamental Theorem of Calculus 1]: ![\displaystyle \int\limits^b_a {f(x)} \, dx = F(b) - F(a)](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cint%5Climits%5Eb_a%20%7Bf%28x%29%7D%20%5C%2C%20dx%20%3D%20F%28b%29%20-%20F%28a%29)
Integration Property [Multiplied Constant]: ![\displaystyle \int {cf(x)} \, dx = c \int {f(x)} \, dx](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cint%20%7Bcf%28x%29%7D%20%5C%2C%20dx%20%3D%20c%20%5Cint%20%7Bf%28x%29%7D%20%5C%2C%20dx)
Integration Property [Addition/Subtraction]: ![\displaystyle \int {[f(x) \pm g(x)]} \, dx = \int {f(x)} \, dx \pm \int {g(x)} \, dx](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cint%20%7B%5Bf%28x%29%20%5Cpm%20g%28x%29%5D%7D%20%5C%2C%20dx%20%3D%20%5Cint%20%7Bf%28x%29%7D%20%5C%2C%20dx%20%5Cpm%20%5Cint%20%7Bg%28x%29%7D%20%5C%2C%20dx)
Integration Property [Splitting Integral]: ![\displaystyle \int\limits^c_a {f(x)} \, dx = \int\limits^b_a {f(x)} \, dx + \int\limits^c_b {f(x)} \, dx](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cint%5Climits%5Ec_a%20%7Bf%28x%29%7D%20%5C%2C%20dx%20%3D%20%5Cint%5Climits%5Eb_a%20%7Bf%28x%29%7D%20%5C%2C%20dx%20%2B%20%5Cint%5Climits%5Ec_b%20%7Bf%28x%29%7D%20%5C%2C%20dx)
Step-by-step explanation:
<u>Step 1: Define</u>
<em>Identify.</em>
![\displaystyle f(x) = \left \{ {{9x^9 ,\ 0 \leq x \leq 1} \atop {4x^3 ,\ 1 \leq x \leq 2}} \right.](https://tex.z-dn.net/?f=%5Cdisplaystyle%20f%28x%29%20%3D%20%5Cleft%20%5C%7B%20%7B%7B9x%5E9%20%2C%5C%200%20%5Cleq%20x%20%5Cleq%201%7D%20%5Catop%20%7B4x%5E3%20%2C%5C%201%20%5Cleq%20x%20%5Cleq%202%7D%7D%20%5Cright.)
![\displaystyle \int\limits^2_0 {f(x)} \, dx = \ ?](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cint%5Climits%5E2_0%20%7Bf%28x%29%7D%20%5C%2C%20dx%20%3D%20%5C%20%3F)
<u>Step 2: Integrate</u>
- [Integral] Rewrite [Integration Property - Splitting Integral]:
![\displaystyle \int\limits^2_0 {f(x)} \, dx = \int\limits^1_0 {f(x)} \, dx + \int\limits^2_1 {f(x)} \, dx](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cint%5Climits%5E2_0%20%7Bf%28x%29%7D%20%5C%2C%20dx%20%3D%20%5Cint%5Climits%5E1_0%20%7Bf%28x%29%7D%20%5C%2C%20dx%20%2B%20%5Cint%5Climits%5E2_1%20%7Bf%28x%29%7D%20%5C%2C%20dx)
- [Integrand] Substitute in function:
![\displaystyle \int\limits^2_0 {f(x)} \, dx = \int\limits^1_0 {9x^9} \, dx + \int\limits^2_1 {4x^3} \, dx](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cint%5Climits%5E2_0%20%7Bf%28x%29%7D%20%5C%2C%20dx%20%3D%20%5Cint%5Climits%5E1_0%20%7B9x%5E9%7D%20%5C%2C%20dx%20%2B%20%5Cint%5Climits%5E2_1%20%7B4x%5E3%7D%20%5C%2C%20dx)
- [Integrals] Rewrite [Integration Property - Multiplied Constant]:
![\displaystyle \int\limits^2_0 {f(x)} \, dx = 9 \int\limits^1_0 {x^9} \, dx + 4 \int\limits^2_1 {x^3} \, dx](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cint%5Climits%5E2_0%20%7Bf%28x%29%7D%20%5C%2C%20dx%20%3D%209%20%5Cint%5Climits%5E1_0%20%7Bx%5E9%7D%20%5C%2C%20dx%20%2B%204%20%5Cint%5Climits%5E2_1%20%7Bx%5E3%7D%20%5C%2C%20dx)
- [Integrals] Integration Rule [Reverse Power Rule]:
![\displaystyle \int\limits^2_0 {f(x)} \, dx = 9 \bigg( \frac{x^{10}}{10} \bigg) \bigg| \limits^1_0 + 4 \bigg( \frac{x^4}{4} \bigg) \bigg| \limits^2_1](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cint%5Climits%5E2_0%20%7Bf%28x%29%7D%20%5C%2C%20dx%20%3D%209%20%5Cbigg%28%20%5Cfrac%7Bx%5E%7B10%7D%7D%7B10%7D%20%5Cbigg%29%20%5Cbigg%7C%20%5Climits%5E1_0%20%2B%204%20%5Cbigg%28%20%5Cfrac%7Bx%5E4%7D%7B4%7D%20%5Cbigg%29%20%5Cbigg%7C%20%5Climits%5E2_1)
- Integration Rule [Fundamental Theorem of Calculus 1]:
![\displaystyle \int\limits^2_0 {f(x)} \, dx = 9 \bigg( \frac{1}{10} \bigg) + 4 \bigg( \frac{15}{4} \bigg)](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cint%5Climits%5E2_0%20%7Bf%28x%29%7D%20%5C%2C%20dx%20%3D%209%20%5Cbigg%28%20%5Cfrac%7B1%7D%7B10%7D%20%5Cbigg%29%20%2B%204%20%5Cbigg%28%20%5Cfrac%7B15%7D%7B4%7D%20%5Cbigg%29)
- Simplify:
![\displaystyle \int\limits^2_0 {f(x)} \, dx = \frac{159}{10}](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cint%5Climits%5E2_0%20%7Bf%28x%29%7D%20%5C%2C%20dx%20%3D%20%5Cfrac%7B159%7D%7B10%7D)
Topic: AP Calculus AB/BC (Calculus I/I + II)
Unit: Integration
Step-by-step explanation:
Hi there!
The best way to approach this problem is Soh Cah Toa.
In this case you can use sine. sine is opposite over hypotenuse.
so sin(17)= b/15
then to solve for b you multiply 15 on both sides to get 4.4