Singapore Airlines is looking to purchase several planes from Boeing. The airline would like to purchase two models of airplanes: the 787-10 Dreamliner built in North Charleston, South Carolina and the 777-9 built in Everett, Washington. Boeing’s 787-10 Dreamliner costs $313 million and the 777-9 costs $408.8 million. Suppose Singapore Airlines negotiates a 12% discount off the final cost.
Let d represent the number of 787-10 Dreamliners the airline would like to purchase and e represent the number of 777-9’s it would like to purchase.
Which of the following statements are true? Select all that apply.
Option 1: The expression 313d represents the cost, in millions, of buying only 787-10 Dreamliners.
Option 2: The expression 408.8e − 0.12(e − d) represents the cost, in millions, of buying only 777-9’s.
Option 3: The expression 0.88(313d + 408.8e) represents the total cost, in millions, of buying both models.
Option 4: If the airline has a budget of $12.1 billion, it can purchase 21 787-10 Dreamliners and 17 777-9’s.
Option 5: If the airline has a budget of $15.3 billion, it can purchase 30 787-10 Dreamliners and 20 777-9’s.
To determine which options are true, let's evaluate each one of them:
Option 1: The expression 313d represents the cost, in millions, of buying only 787-10 Dreamliners.
- True. If "d" represents the number of 787-10 Dreamliners, then 313d would indeed represent the total cost in millions of dollars for "d" Dreamliners before any discounts.
Option 2: The expression 408.8e − 0.12(e − d) represents the cost, in millions, of buying only 777-9’s.
- False. The expression given does not correctly account for a 12% discount nor does it properly represent the total cost of the 777-9s. For the 777-9s, the cost without discount would be 408.8e and with a 12% discount it would be 408.8e - 0.12 * 408.8e, which simplifies to 0.88 * 408.8e.
Option 3: The expression 0.88(313d + 408.8e) represents the total cost, in millions, of buying both models.
- True. If Singapore Airlines gets a 12% discount, they would pay 88% (which is 100% - 12%) of the total cost. Thus, 0.88(313d + 408.8e) correctly calculates the discounted cost in millions of dollars for buying both models.
Option 4: If the airline has a budget of $12.1 billion, it can purchase 21 787-10 Dreamliners and 17 777-9’s.
- To verify this, first calculate the total cost for 21 787-10 Dreamliners and 17 777-9’s without discount and then apply the discount:
Total cost without discount = (21 * 313) + (17 * 408.8) = 6573 + 6949.6 = 13522.6 million dollars
Total cost with discount = 0.88 * 13522.6 = 11899.888 million dollars
Since $11,899.888 million is less than $12,100 million, the airline can indeed purchase 21 Dreamliners and 17 777-9’s with their $12.1 billion budget; this option is true.
Option 5: If the airline has a budget of $15.3 billion, it can purchase 30 787-10 Dreamliners and 20 777-9’s.
- Again, calculate the total cost for 30 787-10 Dreamliners and 20 777-9’s without discount and then apply the discount:
Total cost without discount = (30 * 313) + (20 * 408.8) = 9390 + 8176 = 17566 million dollars
Total cost with discount = 0.88 * 17566 = 15458.08 million dollars
Since $15,458.08 million is more than the $15,300 million budget, the airline cannot purchase 30 Dreamliners and 20 777-9’s with their $15.3 billion budget; this option is false.
In summary, Options 1 and 3 are true, while Options 2, 4 and 5 are false.