Question Video: Finding the Maximum Value of a Function Involving a Rational Function Using the Quotient Rule Mathematics

Video Transcript

Given that the volume of a hot-air balloon grows according to the relation 𝑓 of 𝑡 equals 7000𝑡 over 𝑡 squared plus 49 plus 4000, where the time is measured in hours, determine its maximum volume.

We’ve been given an equation then that governs the growth of this hot-air balloon. And we’re asked to determine its maximum volume. Well, this will take place when the volume of the hot-air balloon is no longer growing, which means that the rate of change of this volume with respect to time will be equal to zero. We know that the rate of change of a quantity is given by its first derivative. So in order to determine the maximum volume, we first need to determine the time at which 𝑓 prime of 𝑡, the first derivative of this function, is equal to zero.

We can use differentiation in order to find an expression for 𝑓 prime of 𝑡. Looking at our function 𝑓 of 𝑡, we see that it is a quotient of two differentiable functions plus a constant. So we’re going to need to use the quotient rule to help us differentiate. The quotient rule tells us that, for two differentiable functions 𝑢 and 𝑣, the derivative of their quotient, 𝑢 over 𝑣, is equal to 𝑣𝑢 prime minus 𝑢𝑣 prime all over 𝑣 squared.

We’re therefore going to let 𝑢 equal the function in the numerator of the quotient, that’s 7000𝑡, and 𝑣 equal the function in the denominator. That’s 𝑡 squared plus 49. We need to find each of their individual derivatives with respect to 𝑡, which we can do using the power rule of differentiation. The derivative of 𝑢 with respect to 𝑡, 𝑢 prime, is equal to 7000. And the derivative of to 𝑣 with respect to 𝑡, 𝑣 prime, is equal to two 𝑡.

Now, we can substitute into the formula for the quotient rule. 𝑓 prime of 𝑡 is equal to 𝑣𝑢 prime, that’s 𝑡 squared plus 49 multiplied by 7000, minus 𝑢𝑣 prime. That’s 7000𝑡 multiplied by two 𝑡. And this is all divided by 𝑣 squared. That’s 𝑡 squared plus 49 all squared. Now, remember, this just gives the derivative of the first part of our function 𝑓 of 𝑡. We also need to remember to differentiate that constant of 4000. But as the derivative of a constant is just zero, it actually makes no contribution to our derivative. We can then distribute the parentheses in the numerator and collect like terms, to give that 𝑓 prime of 𝑡 is equal to 343000 minus 7000𝑡 squared over 𝑡 squared plus 49 squared.

Now, remember, we said that the maximum volume of this hot-air balloon would be achieved when its rate of change with respect to time is equal to zero. So next, we’re going to set our expression for the first derivative equal to zero and solve for 𝑡. We have the equation then, 343000 minus 7000𝑡 squared over 𝑡 squared plus 49 squared is equal to zero. Now, for a quotient to be equal to zero, it must be true that the numerator of the quotient is equal to zero. So actually, we can simplify our equation. We’re just left with 343000 minus 7000𝑡 squared is equal to zero. Adding 7000𝑡 squared to each side and then dividing by 7000 gives 𝑡 squared is equal to 49.

We can solve this equation by square rooting. And as 𝑡 represents time, we’re only going to take the positive value. The positive square root of 49 is seven. So we have that the maximum volume of the hot-air balloon will be achieved when 𝑡 equals seven. That’s after seven hours. Next, we need to determine what the maximum volume actually is. And we can do this by substituting 𝑡 equals seven into our equation 𝑓 of 𝑡. We obtain 7000 multiplied by seven over seven squared plus 49 plus 4000. This is equal to 49000 over 98 plus 4000, which is all equal to 4500. We found then that the maximum volume of the hot-air balloon is 4500 cubic units.

Now, we only found one value for 𝑡 when solving 𝑓 prime of 𝑡 is equal to zero. So this is the only critical point of our function 𝑓 of 𝑡. But we should confirm that it is indeed a maximum. And we can do this using the first derivative test. What we’re going to do is consider the shape of the curve around this critical point by evaluating the slope of the curve. Now, a critical point occurred when 𝑡 was equal to seven. So we’ll choose values of 𝑡 either side of this. Let’s choose six and eight. We know that when 𝑡 is equal to seven, the first derivative in the slope is equal to zero. When 𝑡 is equal to six, the first derivative is equal to 343000 minus 7000 multiplied by six squared over six squared plus 49 squared, which is equal to 12.595. So we find that the slope of the curve is positive when 𝑡 equals six.

On the other side of the critical point, when 𝑡 is equal to eight, the first derivative is equal to 343000 minus 7000 multiplied by eight squared over eight squared plus 49 squared, which is equal to negative 8.223. So the slope of the curve is negative on this side of the critical point. So the slope changes from positive to zero to negative. And so we can see from a sketch of this that the critical point at 𝑡 equals seven is indeed a local maximum.

We’ve completed the problem then. We found the maximum volume of the hot-air balloon to be 4500 cubic units. And using the first derivative test, we’ve confirmed that it is indeed a local maximum.