The rate of change of the angle of elevation when the firework is 40 feet above the ground is 0.12 radians/second.
First we will draw a right angle triangle ΔABC, where ∠B = 90°
Lets, assume the height(AB) = h and base(BC)= x
If the angle of elevation, ∠ACB = α, then
tan(α) = 
Taking inverse trigonometric function, α = tan⁻¹ (
) .............(1)
As we need to find the rate of change of the angle of elevation, so we will differentiate both sides of equation (1) with respect to time (t) :
![\frac{d\alpha}{dt}=[\frac{1}{1+ \frac{h^2}{x^2}}]*(\frac{1}{x})\frac{dh}{dt}](https://tex.z-dn.net/?f=%20%5Cfrac%7Bd%5Calpha%7D%7Bdt%7D%3D%5B%5Cfrac%7B1%7D%7B1%2B%20%5Cfrac%7Bh%5E2%7D%7Bx%5E2%7D%7D%5D%2A%28%5Cfrac%7B1%7D%7Bx%7D%29%5Cfrac%7Bdh%7D%7Bdt%7D%20%20%20%20%20)
Here, the firework is launched from point B at the rate of 10 feet/second and when it is 40 feet above the ground it reaches point A,
that means h = 40 feet and
= 10 feet/second.
C is the observer's position which is 50 feet away from the point B, so x = 50 feet.
![\frac{d\alpha}{dt}= [\frac{1}{1+ \frac{40^2}{50^2}}] *\frac{1}{50} *10\\ \\ \frac{d\alpha}{dt} = [\frac{1}{1+\frac{16}{25}}] *\frac{1}{5}\\ \\ \frac{d\alpha}{dt} = [\frac{25}{41}] *\frac{1}{5}\\ \\ \frac{d\alpha}{dt}= \frac{5}{41} =0.1219512](https://tex.z-dn.net/?f=%20%5Cfrac%7Bd%5Calpha%7D%7Bdt%7D%3D%20%5B%5Cfrac%7B1%7D%7B1%2B%20%5Cfrac%7B40%5E2%7D%7B50%5E2%7D%7D%5D%20%2A%5Cfrac%7B1%7D%7B50%7D%20%2A10%5C%5C%20%5C%5C%20%5Cfrac%7Bd%5Calpha%7D%7Bdt%7D%20%3D%20%5B%5Cfrac%7B1%7D%7B1%2B%5Cfrac%7B16%7D%7B25%7D%7D%5D%20%2A%5Cfrac%7B1%7D%7B5%7D%5C%5C%20%5C%5C%20%5Cfrac%7Bd%5Calpha%7D%7Bdt%7D%20%3D%20%5B%5Cfrac%7B25%7D%7B41%7D%5D%20%2A%5Cfrac%7B1%7D%7B5%7D%5C%5C%20%20%20%5C%5C%20%5Cfrac%7Bd%5Calpha%7D%7Bdt%7D%3D%20%5Cfrac%7B5%7D%7B41%7D%20%3D0.1219512%20%20%20)
= 0.12 (Rounding up to two decimal places)
So, the rate of change of the angle of elevation is 0.12 radians/second.
The greatest common factor (GCF) of 24, 44, and 52 is: 4
Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24
Factors of 44: 1, 2, 4, 11, 22
Factors of 52: 1, 2, 4, 13
The common factors between them are 1, 2, and 4, but 4 is the greatest, making it the GCF.
Answer:
y=3x+19.
Step-by-step explanation:
1) according to the condition the line 't' is parallel to the line 's', it means, the equation of the line 't' is: y=3x+C, where C - unknown number.
2) in order to calculate the 'C', it need to substitute the given coordinates into the equation of the line 't', then:
-2=3*(-7)+C, ⇒ C=19.
3) if C=19 and the equation of the line 't' was y=3x+C, then it is possible to make up the full required equation:
y=3x+19.
Answer:
mean: 30
standard dev: 5
the data point 37 is more than
Step-by-step explanation: