Sarah makes bracelets and necklaces to sell at a craft store. Each bracelet makes a profit of $12, takes 1 hour to assemble, and costs $2 for materials. Each necklace makes a profit of $7, takes 2 hours to assemble, and costs $3 for materials. Sarah has 48 hours available to assemble bracelets and necklaces. (𝑥=number of bracelets, 𝑦=number of necklaces).Maximize 𝑍=12𝑥+7𝑦Constraints: 𝑥≥0,𝑦≥0 2𝑥+3𝑦≤78 𝑥+2𝑦≤48 21. If she has $78 available to
Question
Sarah makes bracelets and necklaces to sell at a craft store. Each bracelet makes a profit of 2 for materials. Each necklace makes a profit of 3 for materials. Sarah has 48 hours available to assemble bracelets and necklaces. (𝑥=number of bracelets, 𝑦=number of necklaces).Maximize 𝑍=12𝑥+7𝑦Constraints: 𝑥≥0,𝑦≥0 2𝑥+3𝑦≤78 𝑥+2𝑦≤48 21. If she has $78 available to
Solution
This is a linear programming problem. We are trying to maximize the profit function Z = 12x + 7y (where x is the number of bracelets and y is the number of necklaces) subject to the constraints:
- x ≥ 0 (She cannot make a negative number of bracelets)
- y ≥ 0 (She cannot make a negative number of necklaces)
- 2x + 3y ≤ 78 (The cost of materials for the bracelets and necklaces cannot exceed $78)
- x + 2y ≤ 48 (The time to make the bracelets and necklaces cannot exceed 48 hours)
To solve this problem, we first graph the feasible region defined by the constraints. Then we find the vertices of this region. The vertices are typically where the maximum or minimum values of the objective function occur in linear programming problems.
The constraints x ≥ 0 and y ≥ 0 give us the first quadrant of the xy-plane. The inequality 2x + 3y ≤ 78 gives a line with a negative slope when rearranged to y ≤ (78 - 2x) / 3, and the region below this line. The inequality x + 2y ≤ 48 gives a line with a negative slope when rearranged to y ≤ (48 - x) / 2, and the region below this line.
The feasible region is the area that satisfies all these conditions. The vertices of this region can be found by solving the system of equations formed by the intersection of the lines 2x + 3y = 78 and x + 2y = 48.
Once we have the vertices, we substitute these into the profit function Z = 12x + 7y to find which gives the maximum profit. This will give us the number of bracelets and necklaces Sarah should make to maximize her profit.
Similar Questions
Sarah makes bracelets and necklaces to sell at a craft store. Each bracelet makes a profit of $7, takes 1 hour to assemble, and costs $2 for materials. Each necklace makes a profit of $12, takes 2 hour to assemble, and costs $3 for materials. Sarah has 48 hours available to assemble bracelets and necklaces. (𝑥=number of bracelets, 𝑦=number of necklaces).Maximize 𝑍=12𝑥+7𝑦Constraints: 𝑥≥0,𝑦≥0 2𝑥+3𝑦≤78 𝑥+2𝑦≤48 22. How much optimum profit did she realize?Group of answer choicesP 468P 270P 300P 168
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opportunity cost of making a necklace
Marisol is making bracelets and rings to sell at a craft fair. She plans to sell each bracelet for $6 and each ring for $8. The craft fair committee charges a $25 fee to sell at the fair, and it costs Marisol $2 to make a bracelet and $4 to make a ring. If Marisol wants to sell at least $600 in jewelry and spend less than $300 for supplies and the fee, which system of inequalities represents the situation? Let b represent the number of bracelets and r represent the number of rings.
The artisans at Jewellery Junction in Phoenix are preparing to make gold jewellery during a 2-month period for the Christmas season. They can make bracelets, necklaces, and pins. Each bracelet requires 6.3 ounces of gold and 17 hours of labour, each necklace requires 3.9 ounces of gold and 10 hours of labour, and each pin requires 3.1 ounces of gold and 7 hours of labour. Jewellery Junction has available 125 ounces of gold and 320 hours of labour. A bracelet sells for $1,650, a necklace for $850, and a pin for $790. How many of each item should be produced to maximize revenue? Ignore the fact that number of items must be integer. A. The antisans should make 13.60 units of bracelets, 12.67 units of necklaces and no pins B. The antisans should make no bracelets, 13.60 units of necklaces and 12.67 units of pins C. The antisans should make 13.60 units of bracelets, no necklaces and 12.67 units of pins D. The antisans should make 12.67 units of bracelets, no necklaces and 13.60 units of pins
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