Antonio's toy boat is bobbing in the water next to a dock. Antonio starts his stopwatch, and measures the vertical distance from
the dock to the height of the boat's mast, which varies in a periodic way that can be modeled approximately by a trigonometric function. The vertical distance from the dock to the boat's mast reaches its highest value of -27 \text{ cm}−27 cmminus, 27, space, c, m every 333 seconds. The first time it reaches its highest point is after 1.31.31, point, 3 seconds. Its lowest value is -44\text{ cm}−44 cmminus, 44, space, c, m. Find the formula of the trigonometric function that models the vertical height HHH between the dock and the boat's mast ttt seconds after Antonio starts his stopwatch. Define the function using radians.
1 answer:
Answer:
Step-by-step explanation:
Since we're given a time at which the height is maximum, we can use a cosine function for the model.
The amplitude is half the difference between the maximum and minimum: (-27 -(-44))/2 = 8.5 cm.
The mean value of the height is the average of the maximum and minimum: (-27 -44)/2 = -35.5 cm.
The period is given as 3 seconds, and the right shift is given as 1.31 seconds.
This gives us enough information to write the function as ...
H(t) = (amplitude)×cos(2π(t -right shift)/period) + (mean height)
H(t) = 8.5cos(2π(t -1.31)/3) -35.5 . . . . cm
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(a) (i) 3.7 billion
(ii) $ 4.6 billion
(b) $ 0.3 billion/year
Step-by-step explanation:
Given equation that shows the amount( in billions of dollar ) spent by consumers on streamed music after t years since 2012,
f(t) = 3.7 + 0.84 ln(t + 1) ,
(a)
(i) for 2012, t = 0,
So, the amount spent by consumers on streamed music in 2012,
f(0) = 3.7 + 0.84 ln (1)
= 3.7 + 0.84(0)
= $ 3.7 billion
(ii) for 2014, t = 2 years,
So, the amount spent by consumers on streamed music in 2014,
f(0) = 3.7 + 0.84 ln (1+2)
= 3.7 + 0.84 ln(3)
= 3.7 + 0.9228
= 4.6228 billion
≈ $ 4.6 billion
(b) ∵ f(t) = 3.7 + 0.84 ln(t + 1),
Differentiating with respect to t,
For t = 2 years,
Hence, the amount spent by consumers on streamed music is growing with the rate of $ 0.3 billion/year in 2014
Answer:
The repulsive force is .
Step-by-step explanation:
Consider the provided information.
The coulomb's law to calculate the repulsive force:
Where the value of k is 9.00×10⁹ Nm²/C²
Substitute the respective values in the above formula.
Hence, the repulsive force is .