Chapter 4 | Applications of Derivatives
445
6− x 3 (6− x ) 2 +4 .
1 8 =
Therefore,
(4.6)
3 (6− x ) 2 +4=8(6− x ). Squaring both sides of this equation, we see that if x satisfies this equation, then x must satisfy 9 ⎡ ⎣ (6− x ) 2 +4 ⎤ ⎦ =64(6− x ) 2 , which implies 55(6− x ) 2 =36. We conclude that if x is a critical point, then x satisfies ( x −6) 2 = 36 55 . Therefore, the possibilities for critical points are x =6± 6 55 .
4.33 Suppose the island is 1 mi from shore, and the distance from the cabin to the point on the shore closest to the island is 15mi. Suppose a visitor swims at the rate of 2.5mph and runs at a rate of 6mph. Let x denote the distance the visitor will run before swimming, and find a function for the time it takes the visitor to get from the cabin to the island. In business, companies are interested in maximizing revenue. In the following example, we consider a scenario in which a company has collected data on how many cars it is able to lease, depending on the price it charges its customers to rent a car. Let’s use these data to determine the price the company should charge to maximize the amount of money it brings in. Example 4.35 Maximizing Revenue Since x =6+6/ 55 is not in the domain, it is not a possibility for a critical point. On the other hand, x =6−6/ 55 is in the domain. Since we squared both sides of Equation 4.6 to arrive at the possible critical points, it remains to verify that x =6−6/ 55 satisfies Equation 4.6 . Since x =6−6/ 55 does satisfy that equation, we conclude that x =6−6/ 55 is a critical point, and it is the only one. To justify that the time is minimized for this value of x , we just need to check the values of T ( x ) at the endpoints x =0 and x =6, and compare them with the value of T ( x ) at the critical point x =6−6/ 55. We find that T (0) ≈ 2.108 h and T (6) ≈ 1.417 h, whereas T ⎛ ⎝ 6−6/ 55 ⎞ ⎠ ≈1.368h. Therefore, we conclude that T has a local minimum at x ≈5.19 mi.
Owners of a car rental company have determined that if they charge customers p dollars per day to rent a car, where 50≤ p ≤200, the number of cars n they rent per day can be modeled by the linear function
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