Flying into a headwind The velocity (in mi/hr) of an airplane ...

Question

Flying into a headwind The velocity (in mi/hr) of an airplane flying into a headwind is given by v(t) = 30(16−t²), for 0≤t≤3. Assume s(0)=0 and t is measured in hours.

c. How far has the airplane traveled at the instant its velocity reaches 400 mi/hr?

Accepted Answer

Welcome back, everyone. A boat's velocity in MPH against a current is given by V of T equals 18 multiplied by 25 minus T squad for time between 0 and 5 inclusive where T is in hours. How far has the boat traveled at the instant? Its velocity first reaches 324 MPH. So first of all, we're going to identify. The time value at which velocity is equal to 324 MPH. We're going to solve the of, let's call it capital T. is equal to 324, and we're going to use our velocity function. So we're going to set 18, multiplied by 25 minus T2 equals 324. So let's divide both sides by 18, we get 25 minus T2 is equal to 18, and therefore T2 is equal to 25 minus 18, which is 7. So now we sold for T. We got two solutions, positive or negative square root of 7, because square root of T squared is the absolute value of T, but we know that our time. Must be greater than or equal to 0, so we only have one solution, right? Capital T is equal to a square root of 7. Now, what we're going to do is simply identify displacement because it says how far, right? So we're going to identify displacement D, let's recall that it is equal to a definite integral between two time values. Specifically, we are going to begin at 0 and end at square root of 7, right? The time value that we have just identified. And we want to integrate the velocity function A multiplied by 25 minus T2 DT. What we can do is simply factor out the constant, which is 18. And then we can integrate term by term. The integral of 25 DT is 25T, and the integral of negative T squared is negative T cubed divided by 3. Using the power rule. We want to evaluate the result from 0 up to square root of 7. Let's apply the fundamental theorem of calculus part 2, we get 18 imprecies. 25 square root of 7. Minus Square root of 7 cubed. Divided by 3. Minus The value of this function adds 0, which is 0, right? Because every term contains 0. So we are ready to simplify. Let's go ahead and do that. We got 18. Multiplied by 25 square root of 7. -18 multiplied by a square root of 7 cubs divided by 3. We can also find the common denominator, which gives us 18 multiplied by. The common denominator would be 3. And our input NC 25 multiplied by 3 is 75, so we get 75 square root of 7 minus. Square root of 7 cubed, that would be 7 square root of 7. So we're multiplying 18 by 68 square root of 7. Divided by 3. We can simplify because 18 divided by 3 is 6, and we get 6 multiplied by 68 square root of 7. Which is equal to 408 square root of 7, and our units will be miles, right? Because our units were given in MPH for velocity. Well done, thank you for watching.

Flying into a headwind The velocity (in mi/hr) of an airplane flying into a headwind is given by v(t) = 30(16−t²), for 0≤t≤3. Assume s(0)=0 and t is measured in hours.c. How far has the airplane traveled at the instant its velocity reaches 400 mi/hr?

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Flying into a headwind The velocity (in mi/hr) of an airplane flying into a headwind is given by v(t) = 30(16−t²), for 0≤t≤3. Assume s(0)=0 and t is measured in hours.

c. How far has the airplane traveled at the instant its velocity reaches 400 mi/hr?