at the beginning of january, kesia records paid $148,950 to acquire the exclusive rights to a new album. it…

at the beginning of january, kesia records paid $148,950 to acquire the exclusive rights to a new album. it costs them $1.13 to print a copy of this album, which they can sell for $9.75. the following chart shows the sales of that record, along with the overhead expenses of running a record studio, not counting production costs.\n| month | albums sold | expenses |\n| ---- | ---- | ---- |\n| jan. | 5,486 | $27,714 |\n| feb. | 8,191 | $21,689 |\n| mar. | 4,796 | $25,195 |\n| apr. | 7,490 | $28,766 |\n| may | 6,272 | $24,604 |\n| jun. | 5,131 | $29,040 |\nin which month did kesia records first break even?\n a. january\n b. march\n c. april\n d. may

at the beginning of january, kesia records paid $148,950 to acquire the exclusive rights to a new album. it costs them $1.13 to print a copy of this album, which they can sell for $9.75. the following chart shows the sales of that record, along with the overhead expenses of running a record studio, not counting production costs.\n| month | albums sold | expenses |\n| ---- | ---- | ---- |\n| jan. | 5,486 | $27,714 |\n| feb. | 8,191 | $21,689 |\n| mar. | 4,796 | $25,195 |\n| apr. | 7,490 | $28,766 |\n| may | 6,272 | $24,604 |\n| jun. | 5,131 | $29,040 |\nin which month did kesia records first break even?\n a. january\n b. march\n c. april\n d. may

Answer

Explanation:

Step1: Calculate profit formula

Let $n$ be the number of albums sold. The cost function $C(n)=148950 + 1.13n+\text{over - head expenses}$. The revenue function $R(n)=9.75n$. The break - even point is when $R(n)=C(n)$, or profit $P(n)=R(n)-C(n)=0$. So $P(n)=9.75n-(148950 + 1.13n+\text{over - head expenses})=8.62n-148950-\text{over - head expenses}$.

Step2: Calculate profit for January

For January, $n = 5486$, overhead expenses $= 27714$. Then $P(5486)=8.62\times5486-148950 - 27714=47299.32-148950 - 27714<0$.

Step3: Calculate profit for February

For February, $n = 8191$, overhead expenses $= 21689$. Then $P(8191)=8.62\times8191-148950 - 21689=70606.42-148950 - 21689<0$.

Step4: Calculate profit for March

For March, $n = 4796$, overhead expenses $= 25195$. Then $P(4796)=8.62\times4796-148950 - 25195=41341.52-148950 - 25195<0$.

Step5: Calculate profit for April

For April, $n = 7490$, overhead expenses $= 28766$. Then $P(7490)=8.62\times7490-148950 - 28766=64563.8-148950 - 28766<0$.

Step6: Calculate profit for May

For May, $n = 6272$, overhead expenses $= 24604$. Then $P(6272)=8.62\times6272-148950 - 24604=54064.64-148950 - 24604<0$.

Step7: Re - check calculations and break - even concept

We made a mistake above. Let's calculate cumulative values. The total cost $C=\sum_{i = 1}^{k}(1.13n_i)+\text{total overhead}+\ 148950$, and the total revenue $R=\sum_{i = 1}^{k}(9.75n_i)$ for $k$ months. Cumulative number of albums sold and overheads:

  • January: $n_1 = 5486$, overhead $h_1=27714$, $R_1=9.75\times5486 = 53498.5$, $C_1=148950+1.13\times5486 + 27714=148950+6209.18+27714=182873.18$
  • February: $n_{1 + 2}=5486 + 8191=13677$, overhead $h_{1+2}=27714 + 21689 = 49403$, $R_{1 + 2}=9.75\times13677=133350.75$, $C_{1 + 2}=148950+1.13\times13677+49403=148950 + 15455.01+49403=213808.01$
  • March: $n_{1+2 + 3}=13677+4796 = 18473$, overhead $h_{1+2+3}=49403+25195 = 74598$, $R_{1+2 + 3}=9.75\times18473=180111.75$, $C_{1+2 + 3}=148950+1.13\times18473+74598=148950+20874.49+74598=244422.49$
  • April: $n_{1+2+3 + 4}=18473+7490=25963$, overhead $h_{1+2+3 + 4}=74598+28766 = 103364$, $R_{1+2+3 + 4}=9.75\times25963=253139.25$, $C_{1+2+3 + 4}=148950+1.13\times25963+103364=148950+29338.19+103364=281652.19$
  • May: $n_{1+2+3+4 + 5}=25963+6272=32235$, overhead $h_{1+2+3+4 + 5}=103364+24604 = 127968$, $R_{1+2+3+4 + 5}=9.75\times32235=314281.25$, $C_{1+2+3+4 + 5}=148950+1.13\times32235+127968=148950+36425.55+127968=313343.55$

Since $R_{1+2+3+4 + 5}>C_{1+2+3+4 + 5}$ in May.

Answer:

d. May