How do you write 5/20 as a percent?????
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Answer: The answers are given below.
Step-by-step explanation: Given that a 3 × 3 matrix P satisfies the matrix equation P² = P.
We are to
(a) find the possibilities for the determinant of P.
(b) explain the reason behind there are no other possibilities.
(c) give an example of P, for each possible determinant.
(a) According to the given information, we have
So, P can be either a zero matrix of order 3 or an identity matrix of order 3.
If P = 0, then det(P) = 0 and if P = I, then det(P) = 1.
Therefore the possible determinants of P are 0 and 1.
(b) There can be any other determinant other than 0 and 1, because if so, then the given equation P² = P will not be satisfied.
(c) If |P| = 0, then the matrix P can be can be of the form as follows:
If |P| = 1, the the matrix P can be of the form as follows :
Thus, all the parts are answered.
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) :
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.
= 0.12 (Rounding up to two decimal places)
So, the rate of change of the angle of elevation is 0.12 radians/second.
