Csc a - sin a = cos a cot a
1/sin a - sin a = cos a (cos a / sin a)
(1 - sin^2 a)/sin a = cos^2 a/sin a
cos^2 a/sin a = cos^2 a/sin a
Therefore, the given equation is an identity.
X=skirt
y=sweater
z=shoes
x+y+z=165
y=8/7x
z=x+11
plug it in
x+(8/7x)+(x+11)=165
2x+8/7x=154
22/7x=154
x=49
y=56
z=60
a skirt cost $49, a sweater cost $56, and a pair of shoes cost $60
This question is solved applying the formula of the area of the rectangle, and finding it's width. To do this, we solve a quadratic equation, and we get that the cardboard has a width of 1.5 feet.
Area of a rectangle:
The area of rectangle of length l and width w is given by:
![A = wl](https://tex.z-dn.net/?f=A%20%3D%20wl)
w(2w + 3) = 9
From this, we get that:
![l = 2w + 3, A = 9](https://tex.z-dn.net/?f=l%20%3D%202w%20%2B%203%2C%20A%20%3D%209)
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:
In this question:
![w(2w+3) = 9](https://tex.z-dn.net/?f=w%282w%2B3%29%20%3D%209)
![2w^2 + 3w - 9 = 0](https://tex.z-dn.net/?f=2w%5E2%20%2B%203w%20-%209%20%3D%200)
Thus a quadratic equation with ![a = 2, b = 3, c = -9](https://tex.z-dn.net/?f=a%20%3D%202%2C%20b%20%3D%203%2C%20c%20%3D%20-9)
Then
![\Delta = 3^2 - 4(2)(-9) = 81](https://tex.z-dn.net/?f=%5CDelta%20%3D%203%5E2%20-%204%282%29%28-9%29%20%3D%2081)
![w_{2} = \frac{-3 - \sqrt{81}}{2*2} = -3](https://tex.z-dn.net/?f=w_%7B2%7D%20%3D%20%5Cfrac%7B-3%20-%20%5Csqrt%7B81%7D%7D%7B2%2A2%7D%20%3D%20-3)
Width is a positive measure, thus, the width of the cardboard is of 1.5 feet.
Another similar problem can be found at brainly.com/question/16995958
Rolling a number less than 4 means that rolling a 1, 2, or 3 will satisfy the requirement. Since 3 of 6 possible outcomes will satisfy the requirement then the likelihood that it will be rolled is 3/6 times. 3/6 reduces to 1/2 or 50% chance.