Kaliska is jumping rope. The vertical height of the center of her rope off the ground R(t) (in cm ) as a function of time t (in seconds) can be modeled by a sinusoidal expression of the form a*cos(b*t)+d. At t=0, when she starts jumping, her rope is 0cm off the ground, which is the minimum. After (pi)/(12) seconds, it reaches a height of 60cm from the ground, which is half of its maximum height. Find R(t). t should be in radians. R(t)=◻

Question

Kaliska is jumping rope. The vertical height of the center of her rope off the ground  R(t) (in  cm ) as a function of time  t (in seconds) can be modeled by a sinusoidal expression of the form  a*cos(b*t)+d. At  t=0, when she starts jumping, her rope is  0cm off the ground, which is the minimum. After  (pi)/(12) seconds, it reaches a height of  60cm from the ground, which is half of its maximum height. Find  R(t).  t should be in radians.  R(t)=◻

Bytelearn · Accepted Answer

Rephrasing the function: Let's first rephrase the  ＂What is the function R(t) that models the vertical height of the center of Kaliska's jump rope off the ground as a function of time?＂Determining the vertical shift: Since the rope starts at 0 cm off the ground at t=0, which is the minimum height, we know that the vertical shift d in the sinusoidal function a*cos(b*t)+d is 0. This is because the cosine function starts at its maximum value when there is no horizontal or vertical shift, and since we're starting at the minimum, the vertical shift must be 0 to flip the cosine wave.Finding the amplitude: Given that the rope reaches half of its maximum height after (pi)/(12) seconds, and this height is 60 cm, we can deduce that the maximum height of the rope is twice this value, which is 120 cm. Therefore, the amplitude a of the sinusoidal function is 120 cm.Calculating the value of b: Now we need to find the value of b, which is related to the period of the function. Since the rope reaches half of its maximum height at (pi)/(12) seconds, and knowing that the cosine function reaches half of its maximum value at pi/3 radians (since cos(pi/3) = 1/2), we can set up the equation b*(pi/12) = pi/3 to solve for b.Solving for b: Solving the equation for b, we get b*(pi/12) = pi/3. Multiplying both sides by 12/pi to isolate b, we get b = (pi/3) * (12/pi) = 4.Writing the function R(t): Now that we have all the parameters for the sinusoidal function, we can write the function R(t) as R(t) = 120*cos(4*t)+0, which simplifies to R(t) = 120*cos(4*t).

Full solution

More problems from Write exponential functions: word problems

Related topics