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.)

Question

UpStudy AI · Accepted 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.
To determine the number of dollars of merchandise that would result in the same total yearly payment under both Plan A and Plan B, set the two cost functions equal: $$ 100 + 0.8x = 40 + 0.9x $$ Solving for $$ x $$: $$ 100 - 40 = 0.9x - 0.8x $$ $$ 60 = 0.1x $$ $$ x = 600 $$ So, purchasing \$600 worth of merchandise in a year results in the same total payment under both plans. For each plan, the total yearly amount paid would be \$580.

Real Tutor Solution

Answer

Solution

Bonus Knowledge

Related Questions

Latest Algebra Questions