Answer: The answer is ![\textup{M}+\textup{H}\leq 10~~\textup{and}4\textup{M}+7\textup{H}\leq 90.](https://tex.z-dn.net/?f=%5Ctextup%7BM%7D%2B%5Ctextup%7BH%7D%5Cleq%2010~~%5Ctextup%7Band%7D4%5Ctextup%7BM%7D%2B7%5Ctextup%7BH%7D%5Cleq%2090.)
Step-by-step explanation: Given that a school wants to put on a dance recital. The programs of Middle school dancer will be short of 4 minutes each and programmes of High school performers will be long of 7 minutes each.
Number of middle school dances is "M" and number od high school dances is "H".
Since there cannot be more than 10 dances all together, so fisrt inequality can be written as
![\textup{M}+\textup{H}\leq 10.](https://tex.z-dn.net/?f=%5Ctextup%7BM%7D%2B%5Ctextup%7BH%7D%5Cleq%2010.)
Also, the entire recital must take maximum 90 minutes, so the second inequality can be written as
![4\textup{M}+7\textup{H}\leq 90.](https://tex.z-dn.net/?f=4%5Ctextup%7BM%7D%2B7%5Ctextup%7BH%7D%5Cleq%2090.)
Thus, the pair of inequalities is
![\textup{M}+\textup{H}\leq 10,\\\\4\textup{M}+7\textup{H}\leq 90.](https://tex.z-dn.net/?f=%5Ctextup%7BM%7D%2B%5Ctextup%7BH%7D%5Cleq%2010%2C%5C%5C%5C%5C4%5Ctextup%7BM%7D%2B7%5Ctextup%7BH%7D%5Cleq%2090.)
Answer:
B) 0.85
Step-by-step explanation:
By definition, complementary angles add to 90 degrees.
The given angle 58 degrees is complementary to 90-58 = 32 degrees. The two angles 58 and 32 add to 90.
Therefore, cos(32) = 0.848 = 0.85
Step-by-step explanation:
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Answer:
The points are: (16.12,0),(-8.12,0).
Step-by-step explanation:
Solving a quadratic equation:
Given a second order polynomial expressed by the following equation:
.
This polynomial has roots
such that
, given by the following formulas:
![x_{1} = \frac{-b + \sqrt{\bigtriangleup}}{2*a}](https://tex.z-dn.net/?f=x_%7B1%7D%20%3D%20%5Cfrac%7B-b%20%2B%20%5Csqrt%7B%5Cbigtriangleup%7D%7D%7B2%2Aa%7D)
![x_{2} = \frac{-b - \sqrt{\bigtriangleup}}{2*a}](https://tex.z-dn.net/?f=x_%7B2%7D%20%3D%20%5Cfrac%7B-b%20-%20%5Csqrt%7B%5Cbigtriangleup%7D%7D%7B2%2Aa%7D)
![\bigtriangleup = b^{2} - 4ac](https://tex.z-dn.net/?f=%5Cbigtriangleup%20%3D%20b%5E%7B2%7D%20-%204ac)
Distance between two points:
Suppose we have two points,
and
. The distance between them is given by:
![D = \sqrt{(x_2-x_1)^2+(y_2-y_1)^2}](https://tex.z-dn.net/?f=D%20%3D%20%5Csqrt%7B%28x_2-x_1%29%5E2%2B%28y_2-y_1%29%5E2%7D)
Find all points on the x-axis that are 14 units from the point (4, -7).
Being on the x-axis mean that they have y-coordinate equal to 0, so the point is (x,0).
The distance is 14. So
![\sqrt{(x-4)^2+(0-(-7))^2} = 14](https://tex.z-dn.net/?f=%5Csqrt%7B%28x-4%29%5E2%2B%280-%28-7%29%29%5E2%7D%20%3D%2014)
![\sqrt{x^2 - 8x + 16 + 49} = 14](https://tex.z-dn.net/?f=%5Csqrt%7Bx%5E2%20-%208x%20%2B%2016%20%2B%2049%7D%20%3D%2014)
![\sqrt{x^2 - 8x + 65} = 14](https://tex.z-dn.net/?f=%5Csqrt%7Bx%5E2%20-%208x%20%2B%2065%7D%20%3D%2014)
![(\sqrt{x^2 - 8x + 65})^2 = 14^2](https://tex.z-dn.net/?f=%28%5Csqrt%7Bx%5E2%20-%208x%20%2B%2065%7D%29%5E2%20%3D%2014%5E2)
![x^2 - 8x + 65 - 196 = 0](https://tex.z-dn.net/?f=x%5E2%20-%208x%20%2B%2065%20-%20196%20%3D%200)
![x^2 - 8x - 131 = 0](https://tex.z-dn.net/?f=x%5E2%20-%208x%20-%20131%20%3D%200)
So ![a = 1, b = -8, c = -131](https://tex.z-dn.net/?f=a%20%3D%201%2C%20b%20%3D%20-8%2C%20c%20%3D%20-131)
![\bigtriangleup = (-8)^{2} - 4(1)(-131) = 588](https://tex.z-dn.net/?f=%5Cbigtriangleup%20%3D%20%28-8%29%5E%7B2%7D%20-%204%281%29%28-131%29%20%3D%20588)
![x_{1} = \frac{-(-8) + \sqrt{588}}{2} = 16.12](https://tex.z-dn.net/?f=x_%7B1%7D%20%3D%20%5Cfrac%7B-%28-8%29%20%2B%20%5Csqrt%7B588%7D%7D%7B2%7D%20%3D%2016.12)
![x_{1} = \frac{-(-8) - \sqrt{588}}{2} = -8.12](https://tex.z-dn.net/?f=x_%7B1%7D%20%3D%20%5Cfrac%7B-%28-8%29%20-%20%5Csqrt%7B588%7D%7D%7B2%7D%20%3D%20-8.12)
The points are: (16.12,0),(-8.12,0).