4. A local club is arranging a charter flight to Hawaii. The cost of the trip is $558 each for 81 passengers, with a refund of $5 per passenger for each passenger in excess of 81. a. Find the number of passengers that will maximize the revenue received from the flight. b. Find the maximum revenue. a. The number of passengers that will maximize the revenue received from the flight is (Round to the nearest integer as needed.) b. The maximum revenue is $ A rectangular tank with a square base, an open top, and a volume of 108 ft° is to be constructed of sheet steel. Find the dimensions of the tank that has the minimum surface area. 5. The dimensions of the tank with minimum surface area are ft. (Simplify your answer. Use a comma to separate answers.) A closed box with a square base is to have a volume of 128,625 cm°. The material for the top and bottom of the box costs $7.50 per square centimeter, while the material for the sides costs $2.50 per square centimeter. Find the dimensions of the box that will lead to the minimum total cost. What is the minimum total cost? 6. Write an equation for C(x), the cost of the box as a function of x, the length of a side of the base. C(x) = The length and width are each (0) (1)- your choice The height is (2) The minimum total cost is $ For he (1) O cm. (2) cm2. For the cm cm?. cm O cm. Al the low At the upper dooint
4. A local club is arranging a charter flight to Hawaii. The cost of the trip is $558 each for 81 passengers, with a refund of $5 per passenger for each passenger in excess of 81. a. Find the number of passengers that will maximize the revenue received from the flight. b. Find the maximum revenue. a. The number of passengers that will maximize the revenue received from the flight is (Round to the nearest integer as needed.) b. The maximum revenue is $ A rectangular tank with a square base, an open top, and a volume of 108 ft° is to be constructed of sheet steel. Find the dimensions of the tank that has the minimum surface area. 5. The dimensions of the tank with minimum surface area are ft. (Simplify your answer. Use a comma to separate answers.) A closed box with a square base is to have a volume of 128,625 cm°. The material for the top and bottom of the box costs $7.50 per square centimeter, while the material for the sides costs $2.50 per square centimeter. Find the dimensions of the box that will lead to the minimum total cost. What is the minimum total cost? 6. Write an equation for C(x), the cost of the box as a function of x, the length of a side of the base. C(x) = The length and width are each (0) (1)- your choice The height is (2) The minimum total cost is $ For he (1) O cm. (2) cm2. For the cm cm?. cm O cm. Al the low At the upper dooint
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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Transcribed Image Text:### Optimization Problems in Mathematics
#### Problem 4: Charter Flight Revenue Maximization
A local club is arranging a charter flight to Hawaii. The cost of the trip is $558 each for 81 passengers, with a refund of $5 per passenger for each passenger in excess of 81.
- **Task**: Determine the optimal number of passengers to maximize revenue and calculate the maximum revenue.
a. Find the number of passengers that will maximize the revenue received from the flight.
**Answer**: The number of passengers that will maximize the revenue received from the flight is __________. (Round to the nearest integer as needed.)
b. Find the maximum revenue.
**Answer**: The maximum revenue is $__________.
---
#### Problem 5: Surface Area Optimization of a Rectangular Tank
A rectangular tank with a square base, an open top, and a volume of 108 cubic feet is to be constructed of sheet steel.
- **Task**: Find the dimensions of the tank that has the minimum surface area.
**Answer**: The dimensions of the tank with minimum surface area are __________ ft. (Simplify your answer. Use a comma to separate answers.)
---
#### Problem 6: Cost Minimization of a Closed Box
A closed box with a square base has to have a volume of 128,625 cubic centimeters. The material for the top and bottom of the box costs $7.50 per square centimeter, while the material for the sides costs $2.50 per square centimeter.
- **Task**: Find the dimensions of the box that will lead to the minimum total cost.
- Write an equation for \(C(x)\), the cost of the box as a function of \(x\), the length of a side of the base.
\(C(x) = __________\)
- Determine:
- The length and width are each ___(1)___ cm.
- The height is ___(2)___ cm.
- The minimum total cost is $__________.
Choices for units:
- cm
- cm\(^3\)
- cm\(^2\)
Each task focuses on optimizing a different mathematical model to find solutions that balance costs, dimensions, and other constraints.
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