The company Jugueton sells two electric ride$-$on cars. They buy all the components from different suppliers and they assemble by hand the products. The ride$-$on cars are tested before being packaged for sale. The difference between the products is that car Y also has a remote$-$control feature. The following budgeted details relate to the ride$-$on cars:

The following costs are derived from their budget this year:

During the following month there is a limitation in the number of hours allocated for assembling and testing because of a legal union action. For assembly, they are limited to $100$ hours and for testing $50$ hours. The managers want to find out how the resources should be distributed by product in order to maximize their profits.

Required

$1.$ Calculate the contribution per unit for each type of product $(5$ p$).$

$2.$ Determine the objective function based on these two constraints $(2$ p$).$

$3.$ Write the equations for the two constraints $(2$ p$)$

$4$ Determine the mix of products that will maximize profits for the business, using a graphical linear programming solution:

$($a$)$ Solve the equations to calculate the points where both lines intercept. $(2\mathrm{.}5$p$)$

$($b$)$ Draw the lines for the constraints and identify the corner points that make up the feasible region. $(5$p$)$

$($c$)$ Determine the mixed of products for the corner point where both lines intercept using the substitution method. $(5$p$)$

$5.$ Identify the mix of products that will maximize the contribution and the total contribution.$(2\mathrm{.}5$p$)$

$6.$ The manager wants to consider the maximum sales that they can achieve for each product. They can sell up to $200$ items of X and $220$ items of Y$.$ Reassess the feasible region and determine whether the mix of products that maximize contribution has changed. Also update the graph. $(6$p$)$

Question $2(10$ points$)$

Explain the concept of multi$-$limiting factor analysis in managerial accounting. How does it differ from analyzing a single limiting factor? Provide an example scenario where a business might encounter multiple limiting factors and discuss how they could prioritize resource allocation in such a situation.

Question $3$: $(10$ points$)$

Discuss the challenges and considerations involved in decision$-$making when a business faces multiple limiting factors.