19.07.2022 - 07:50

# The first year a toy manufacturer introduces a new toy, its sales total $495,000. The company expects its sales to drop 10% each succeeding year. Find the total expected sales in the first 6 yr. Find the total expected sales if the company offers the toy Question: The first year a toy manufacturer introduces a new toy, its sales total {eq}$495,000 {/eq}. The company expects its sales to drop {eq}10 % {/eq} each succeeding year. Find the total expected sales in the first {eq}6 {/eq} yr. Find the total expected sales if the company offers the toy for sale for as long as anyone buys it.

• To find the expected sales in the first 6 years, we can use geometric sequence formula: {eq}a_n = a_1(r)^{n-1} {/eq} where {eq}a_n {/eq} represents the sales in year {eq}n {/eq}, {eq}a_1 {/eq} represents the initial sales, and {eq}r {/eq} represents the common ratio, which is {eq}0.9 {/eq} since the sales drop by 10% each year. Using this formula, we can find the sales in each year: Year 1: {eq}a_1 = $495,000 {/eq} Year 2: {eq}a_2 =$495,000(0.9)^1 \approx $445,500 {/eq} Year 3: {eq}a_3 =$495,000(0.9)^2 \approx $400,950 {/eq} Year 4: {eq}a_4 =$495,000(0.9)^3 \approx $360,855 {/eq} Year 5: {eq}a_5 =$495,000(0.9)^4 \approx $324,769 {/eq} Year 6: {eq}a_6 =$495,000(0.9)^5 \approx 292,292 {/eq} To find the total expected sales in the first 6 years, we can simply add up the sales from each year: {eq}begin{align*} \text{ Total sales} &= a_1 + a_2 + a_3 + a_4 + a_5 + a_6 \ &=495,000 + $445,500 +$400,950 + $360,855 +$324,769 + $292,292 \ &=$2,319,366 end{align*} {/eq} To find the total expected sales if the company offers the toy for sale for as long as anyone buys it, we can use the sum of an infinite geometric series formula: {eq}frac{a_1}{1-r} {/eq} where {eq}a_1 {/eq} and {eq}r {/eq} are the same as before. Using this formula, we can find the total expected sales: {eq}begin{align*} \text{ Total sales} &= \frac{a_1}{1-r} \ &= \frac{$495,000}{1-0.9} \ &= \frac{$495,000}{0.1} \ &= 4,950,000 end{align*} {/eq} Therefore, if the company offers the toy for sale for as long as anyone buys it, the total expected sales would be {eq}4,950,000 {/eq}.