9514 1404 393
Answer:
- A = (0, 1)
- B = (3, -2)
- area = 4.5 square units
Step-by-step explanation:
Rewriting the equations to make x the subject, we have ...
x = y² -1 . . . . . [eq1]
x = 1 - y . . . . . .[eq2]
At the points of intersection, the difference will be zero.
y² -1 -(1 -y) = 0
y² +y -2 = 0
(y -1)(y +2) = 0
The y-coordinates of points A and B are 1 and -2.
The corresponding x-coordinates are ...
x = 1 -{1, -2} = {1 -1, 1+2} = {0, 3}
Then A = (0, 1) and B = (3, -2).
__
A differential of area can be written ...
(x2 -x1)dy = ((1 -y) -(y² -1))dy = (2 -y -y²)dy
Integrating this over the interval y = [-2, 1] gives the area.
![\displaystyle A=\int_{-2}^1(2-y-y^2)\,dy=\left.(2y-\dfrac{1}{2}y^2-\dfrac{1}{3}y^3)\right|_{-2}^1\\\\=\left(2-\dfrac{1}{2}-\dfrac{1}{3}\right)-\left(2(-2)-\dfrac{(-2)^2}{2}-\dfrac{(-2)^3}{3}\right)=\dfrac{7}{6}+4+2-\dfrac{8}{3}\\\\=\boxed{4.5}](https://tex.z-dn.net/?f=%5Cdisplaystyle%20A%3D%5Cint_%7B-2%7D%5E1%282-y-y%5E2%29%5C%2Cdy%3D%5Cleft.%282y-%5Cdfrac%7B1%7D%7B2%7Dy%5E2-%5Cdfrac%7B1%7D%7B3%7Dy%5E3%29%5Cright%7C_%7B-2%7D%5E1%5C%5C%5C%5C%3D%5Cleft%282-%5Cdfrac%7B1%7D%7B2%7D-%5Cdfrac%7B1%7D%7B3%7D%5Cright%29-%5Cleft%282%28-2%29-%5Cdfrac%7B%28-2%29%5E2%7D%7B2%7D-%5Cdfrac%7B%28-2%29%5E3%7D%7B3%7D%5Cright%29%3D%5Cdfrac%7B7%7D%7B6%7D%2B4%2B2-%5Cdfrac%7B8%7D%7B3%7D%5C%5C%5C%5C%3D%5Cboxed%7B4.5%7D)
The area of the shaded region is 4.5 square units.
Answer:
1:10 AM
Step-by-step explanation:
The 4th eruption of the day as at 12:22 PM. And since the time between each eruption gets 12 minutes added, the next eruption is going to happen in 48 minutes, which is at 1:10 AM. After that eruption, the time will reset and the next eruption will happen at 1:22 AM
In order to verify this identity we can expand and simplify the left hand side of the equation. If it becomes equal to the right hand side, this will prove theat the given equation is an identity.
Left hand side = cos (270 - q)
= cos(270)sin(q) + sin(270)cos(q)
cos(270) = -1
sin(270) = 0
So the left hand side becomes
= -1*sin(q) + 0*cos(q)
= - sin(q)
= Right Hand side
This proves the left hand side is equal to the right hand side and so given equation is an identity which means it is true for all values of q.
Use cos,
COS= ADJ/HYP
COS(61)= 5.5/X
Final answer:
X= 5.5/COS(61)
X= 0.1 [rounded to nearest tenth]
Answer:
![y = x + 16](https://tex.z-dn.net/?f=y%20%3D%20x%20%2B%2016)
Step-by-step explanation:
Given
![(x_1,y_1) = (5,21)](https://tex.z-dn.net/?f=%28x_1%2Cy_1%29%20%3D%20%285%2C21%29)
![(x_2,y_2) = (9,25)](https://tex.z-dn.net/?f=%28x_2%2Cy_2%29%20%3D%20%289%2C25%29)
Required
Determine the linear function rule
First, calculate the slope (m)
![m = \frac{y_2 - y_1}{x_2 - x_1}](https://tex.z-dn.net/?f=m%20%3D%20%5Cfrac%7By_2%20-%20y_1%7D%7Bx_2%20-%20x_1%7D)
This gives:
![m = \frac{25 - 21}{9 - 5}](https://tex.z-dn.net/?f=m%20%3D%20%5Cfrac%7B25%20-%2021%7D%7B9%20-%205%7D)
![m = \frac{4}{4}](https://tex.z-dn.net/?f=m%20%3D%20%5Cfrac%7B4%7D%7B4%7D)
![m =1](https://tex.z-dn.net/?f=m%20%3D1)
The equation is then calculated using:
![y = m(x - x_1) + y_1](https://tex.z-dn.net/?f=y%20%3D%20m%28x%20-%20x_1%29%20%2B%20y_1)
![y = 1(x - 5) + 21](https://tex.z-dn.net/?f=y%20%3D%201%28x%20-%205%29%20%2B%2021)
![y = x - 5 + 21](https://tex.z-dn.net/?f=y%20%3D%20x%20-%205%20%2B%2021)
![y = x + 16](https://tex.z-dn.net/?f=y%20%3D%20x%20%2B%2016)