Question 30, 1.10.7 Yourt 3 of 4 Yay are choosing between two plans at a discount warehouse. Plan A offers an annual membership of \( \$ 100 \) and you pores \( 58.76 \%, 61.7 \) of \( 90 \% \) of the manufacturer's list price. a. Express the total yearly amount paid to the warehouse under plan \( \mathrm{A}, \mathrm{f} \), as a function of the dollars of merchandise purchased during the year, x . \( \mathrm{f}(\mathrm{x})=100+0.8 \mathrm{x} \) (Users an annual membership fee of \( \$ 40 \) and you pay b. Express the total yearly amount paid to the warehouse under plan \( \mathrm{B}, \mathrm{g} \), as a function of the dollars of merchandise purchased during the year, x . \( \mathrm{g}(\mathrm{x})=40+0.9 \mathrm{x} \) (Use integers or decimals for any numbers in the expression.) c. How many dollars of merchandise would you have to purchase in a year to pay the same amount under both plans? What will be the total yearly amount paid to the warehouse for each plan? Dollars of merchandise \( =\$ \square \) (Simplify yjour answer.)

To find the number of dollars of merchandise that would need to be purchased in a year to pay the same amount under both plans, we need to set the two functions equal to each other and solve for \( x \). Given: Plan A: \( f(x) = 100 + 0.8x \) Plan B: \( g(x) = 40 + 0.9x \) Setting the two functions equal to each other: \[ 100 + 0.8x = 40 + 0.9x \] Now, we can solve for \( x \) to find the number of dollars of merchandise that would need to be purchased in a year to pay the same amount under both plans. Solve the equation by following steps: - step0: Solve for \(x\): \(100+0.8x=40+0.9x\) - step1: Move the expression to the left side: \(0.8x-0.9x=40-100\) - step2: Add and subtract: \(-0.1x=40-100\) - step3: Add and subtract: \(-0.1x=-60\) - step4: Change the signs: \(0.1x=60\) - step5: Divide both sides: \(\frac{0.1x}{0.1}=\frac{60}{0.1}\) - step6: Divide the numbers: \(x=600\) The number of dollars of merchandise that would need to be purchased in a year to pay the same amount under both plans is \( x = 600 \) dollars. Now, we can substitute \( x = 600 \) into either function to find the total yearly amount paid to the warehouse for each plan. For Plan A: \[ f(600) = 100 + 0.8(600) = 100 + 480 = 580 \] For Plan B: \[ g(600) = 40 + 0.9(600) = 40 + 540 = 580 \] Therefore, the total yearly amount paid to the warehouse for each plan would be $580.