Question

1. roland’s boat tours sells deluxe and economy seats for each tour it conducts. in order to complete a tour, at least 1 economy seats must be sold and at least 6 deluxe seats must be sold. the maximum number of passengers allowed on each boat is 30 roland’s boat tours makes $40 profit for each economy seat sold and $35 profit for each deluxe seat sold. what is the maximum profit from one tour?

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1. the senior class is having a fundraiser to help pay for the senior trip. selling a box of chocolates yields a profit of $2.45, while selling a box of cookies yields a profit of $2.70. the demand for cookies is at least twice that of chocolates, but the amount of cookies produced must be no more than 550 boxes plus 3 times the number of chocolates produced. assuming that the senior class can sell every box that they order, how many boxes of each should they order to maximize profit if they cannot order more than 1950 boxes combined?

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1. a projectile is fired from ground level with an initial velocity of 35 m/s at an angle of 35° with the horizontal. how long will it take for the projectile to reach the ground?

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Explanation:

Problem 1:

Step1: Define Variables

Let \( x \) be the number of economy seats, \( y \) be the number of deluxe seats.
Constraints:

• \( x \geq 1 \)
• \( y \geq 6 \)
• \( x + y \leq 30 \)

Objective function (profit \( P \)): \( P = 40x + 35y \)

Step2: Analyze Profit per Seat

Economy seats give higher profit (\$40) than deluxe (\$35). So we want to maximize \( x \) (economy seats) while satisfying constraints.

Step3: Maximize Economy Seats

Since \( y \geq 6 \), to maximize \( x \), set \( y = 6 \). Then \( x = 30 - 6 = 24 \) (since \( x + y \leq 30 \)).

Step4: Calculate Profit

Substitute \( x = 24 \), \( y = 6 \) into \( P \):
\( P = 40(24) + 35(6) = 960 + 210 = 1170 \)

Step1: Define Variables

Let \( c \) = boxes of chocolates, \( k \) = boxes of cookies.
Constraints:

• \( k \geq 2c \)
• \( k \leq 550 + 3c \)
• \( c + k \leq 1950 \)
• \( c \geq 0, k \geq 0 \)

Objective function (profit \( P \)): \( P = 2.45c + 2.70k \)

Step2: Analyze Profit per Box

Cookies have higher profit (\$2.70) than chocolates (\$2.45). So maximize \( k \) (cookies) while satisfying constraints.

Step3: Solve Constraints

From \( c + k \leq 1950 \) and \( k \geq 2c \), substitute \( k = 2c \) into \( c + k \leq 1950 \):
\( c + 2c \leq 1950 \Rightarrow 3c \leq 1950 \Rightarrow c \leq 650 \)
Then \( k = 2(650) = 1300 \)
Check \( k \leq 550 + 3c \): \( 1300 \leq 550 + 3(650) = 550 + 1950 = 2500 \) (true).

Step4: Calculate Profit

\( P = 2.45(650) + 2.70(1300) = 1592.5 + 3510 = 5102.5 \)

Step1: Vertical Motion Formula

For projectile motion, vertical displacement \( y = v_0 \sin\theta \cdot t - \frac{1}{2}gt^2 \).
Initial velocity \( v_0 = 35 \, \text{m/s} \), angle \( \theta = 35^\circ \), \( g = 9.8 \, \text{m/s}^2 \), \( y = 0 \) (starts and ends at ground level).

Step2: Substitute Values

\( 0 = 35\sin(35^\circ) \cdot t - \frac{1}{2}(9.8)t^2 \)
Factor out \( t \):
\( t(35\sin(35^\circ) - 4.9t) = 0 \)
Solutions: \( t = 0 \) (launch) or \( 35\sin(35^\circ) - 4.9t = 0 \)

Step3: Solve for \( t \)

\( t = \frac{35\sin(35^\circ)}{4.9} \)
\( \sin(35^\circ) \approx 0.5736 \)
\( t \approx \frac{35(0.5736)}{4.9} \approx \frac{20.076}{4.9} \approx 4.1 \, \text{seconds} \)

Answer:

The maximum profit from one tour is \$1170.

Problem 2: