Sharjah Engine Company manufactures and sells diesel engines for use in small farming equipment. For its $2024$ budget, Sharjah Engine Company estimates the following:

Selling price $ $4,000$

Variable cost per engine $ $1,000$

Annual fixed costs $$4,500,000$

Net income $$1,200,000$

Income tax rate $20\%$

The first$-$quarter income statement, as of March $31,$ reported that sales were not meeting expectations. During the first quarter, only $400$ units had been sold at the current price of $$4,000.$ The income statement showed that variable and fixed costs were as planned, which meant that the $2024$ annual net income projection would not be met unless management took action. A management committee was formed and presented the following mutually exclusive alternatives to the president:

Required:

$1.$ Reduce the selling price by $15\%.$ The sales organization forecasts that at this significantly reduced price, $2,500$ units can be sold during the remainder of the year. Total fixed costs and variable cost per unit will stay as budgeted.

$2.$ Lower variable cost per unit by $$300$ through the use of less$-$expensive direct materials. The selling price will also be reduced by $$400,$ and sales of $1,750$ units are expected for the remainder of the year.

$3.$ Reduce fixed costs by $10\%$ and lower the selling price by $30\%.$ Variable cost per unit will be unchanged. Sales of $2,200$ units are expected for the remainder of the year.

a$.$ If no changes are made to the selling price or cost structure, determine the number of units that Sharjah Engine Company must sell $($i$)$ to break even and $($ii$)$ to achieve its net income objective.

b$.$ Determine which alternative Sharjah Engine should select to achieve its net income objective. Show your calculations.