Someone is traveling in the latest SUV by Uber. The car consumes gas at the rate of (1/400) * (1000/x + x) gallons per mile when driven at x miles per hour. The cost of gas is $3.50 per gallon. The driver also has to be paid at the rate of $12.50 per hour. In particular, the question prompts to find the optimal speed that will minimize the total cost in a round trip of 800 miles. The answer choices are 49 Miles Per Hour, 50 Miles Per Hour, 53 Miles Per Hour, 54 Miles Per Hour, and 55 Miles Per Hour. The answer to this particular question is 49 Miles Per Hour.
We first have to start with the gasoline consumption and the cost of gas per mile. The total gasoline consumption equals distance times rate. The distance is 800 miles, and the rate is (1/400) * (1000/x + x). So distance times rate equals (800) * (1/400) * (1000/x + x). Since 800 * (1/400) equals 2, the outcome would be 2 * (1000/x + x). Next, we have to find the total cost. Since the cost per gallon is $3.50, we multiply that by 2 * (1000/x + x). 3.5 times 2 is 7, so we multiply that by (1000/x + x). That would equal 7000/x + 7x. That means the total cost of the gas is 7000/x + 7x. Next, we have to go over the amount of time the doctor drives to figure out the total cost by paying the driver. So if we want to find the time, we just divide 800 by x (the miles per hour). Then you multiply (800/x) with 12.5. 800 multiplied by 12.5 would equal 10000. That would make the final cost 10000/x = the cost for the driver.
In the last paragraph we explained how to get to the total costs of the gasoline consumption and cost for the driver. Now we have to combine the total costs so we can gather up the final costs for each speed. To make the problem easier from 7000/x + 7x + 10000/x we can simplify that to 17000/x + 7 so we can solve this easier. So we can interchange the x’s to 49, 50, 53, 54, and 55. When x = 49, Minimum Total Cost = (17000/49) + (7 * 49) = $680.94. When x = 50, Minimum Total Cost = (17000/50) + (7 * 50) = $690.00. When x = 53, Minimum Total Cost = (17000/53) + (7 * 53) = $691.76. When x = 54, Minimum Total Cost = (17000/54) + (7 * 54) = $692.82. When x = 55, Minimum Total Cost = (17000/55) + (7 * 55) = $694.01. If we look at all the total costs you would see that 49 has the cheapest with $680.94.
To summarize it all, we first have to find the total gasoline consumption using distance times rate. Next, we use that to find the total cost of gas. After that, we have to find the time it takes for the driver to drive 800 miles. We can use that to find the total payment to the driver. Then, we finally add them up and replace the x’s with the variable to find the cheapest number, and we end up with 49 Miles Per Hour. 49 is the best because the minimum total cost of 49 is less expensive than any other price given.