A woman attached to a bungee cord jumps from a bridge that is...

Question

A woman attached to a bungee cord jumps from a bridge that is 30 m above a river. Her height in meters above the river t seconds after the jump is y(t) = 15(1+e^-t cos t), for t ≥ 0.

Determine her velocity at t = 1 and t = 3.

Accepted Answer

Hi everyone. Let's take a look at this practice problem. This problem says a drone released from a building's rooftop starts to ascend vertically with its height above the ground in meters given by the function H of T is equal to 10, multiplied by the quantity of 2 + E ratio of the quantity of minus T in quantity multiplied by sine of 2T, where T is the time in seconds. Determine the velocity of the drone at T equal to 0.5 and T equal to 1 seconds. Round your answers to two decimal places. We're given four possible choices as our answers. For choice A, we have V of 0.5, is approximately 1.45 m per second, and V of 1 is approximately minus 5.28 m per second. For choice B, we have V of 0.5, is approximately 1.45 m per second, and V of 1 is approximately 6.4. 1 m per second. For choice C we have V of 0.5, is approximately 2.32 m per second, and V of 1 is approximately minus 5.28 m per second. And for choice D we have V of 0.5, is approximately 2.32 m per second, and V1 is approximately minus 6.41 m per second. Now asked to determine the velocity of the drone at T equal to 0.5 and T equal to 1 seconds. So, we first need to determine our velocity as a function of time. And so recall that our velocity is given by the time derivative of our position equation. That means our velocity as a function of time, which is VFT. It's going to be equal to the derivative with respect to T of H of T. This H of T is our position equation. So this is going to be equal to the derivative with respect to T of the quantity of 10 multiplied by the quantity of 2, plus e rates to the quantity of minus T and quantity multiplied by sin of 2 T. So taking this derivative, this is going to be equal to 10 here is a constant, so we can move through the derivative um by our constant multiple rule for derivatives. So we'll have 10, multiply in the quantity, and here we'll take a derivative of the remaining terms. So I want to take a derivative of 2 with respect to T, I'll get 0, since I'm taking a derivative of a constant. Then we'll have plus. I'm going to take a derivative of the next term, which is E raised to the quantity of minus T in quantity multiplied by sin of 2 T. I'm going to have to use our product rule. So we call for our product rule that we'll take a derivative of our first term, which in this case is E raised to the minus T. So I want to take its derivative, I'll get minus E raised to the quantity of minus T in quantity. That gets multiplied by the second term in a product, which is sin of 2 T. Then we'll have plus our first term, which is E rated to the quantity of minus T in quantity, multiplied by the derivative of our second term, which is the derivative of sine of 2 T. I want to take it that derivative, I'll get cosine of 2 T, but I'll have to apply a chain rule, so we'll have to multiply this by the derivative of 2T, which in this case is 2. So simplifying this expression, I notice that I can factor out an E raised to the quantity of minus T out of both terms. So I'll have this being equal to 10, multiplied by E raised to the quantity of minus T and quantity, multiplied by the quantity of 2, multiplied by cosine of 2 T minus sine of 2 T. This is our velocity as a function of time. So now we just need to evaluate this expression when T is equal to 0.5 and T is equal to 1. So, we'll start off by calculating the of 0.5. And so this is going to be equal to 10 multiplied by E raised to the quantity of minus 0.5 in quantity, multiplied by the quantity of 2, multiplied by cosine of 2 multiplied by 0.5. Then we'll have minus sine. Of 2 multiplied by 0.5. So, now we just need to evaluate this expression in our calculator. And doing so and just rounding to two decimal places, this is approximately equal to. 1.45 m per second. And so that is the first part of our answer, um, first part of our question answered. And here the units of meters per second come from the fact that our height was given in meters, and our time was given in seconds, and we've taken a derivative of our height with respect to time. So that's the reason why we get units of meters per second. So now we just need to calculate um our velocity when T is equal to 1. So we'll have V of 1. is equal to 10 multiplied by E raised to the minus 1, multiplied by the quantity of 2 multiplied by cosine of 2 multiplied by 1. Minus sign of 2 multiplied by 1. And when we evaluate this expression in our calculator, This turns out to be approximately equal to minus 6.41 m per second. So that's the second half of our question answered. So now that we have both of our values calculated, we can compare them to our answer choices, and we see that the correct answer choice corresponds to answer choice B. So I hope that this explanation has been useful, and I'll see you in the next video.

A woman attached to a bungee cord jumps from a bridge that is 30 m above a river. Her height in meters above the river t seconds after the jump is y(t) = 15(1+e^-t cos t), for t ≥ 0.
Determine her velocity at t = 1 and t = 3.

Key Concepts

Differentiation

Exponential Functions

Trigonometric Functions

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