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1 year ago

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A salesperson makes a base salary of $15,000 per year plus a 8% commission on total sales for the year. The yearly salary can be

expressed as a linear function.

1 answer:

allochka39001 [22]1 year ago

4 0

The salary of the salesperson can be represented as "y" and the number of sales for the year will be represnted as "x". With this in mind the salary will be the fixed part (15000) plus the number of sales per year multiplied by 8 percent. With this in mind we have:

$\begin{gathered} y(x)=15000+8\text{\%}\cdot x \\ y(x)=15000+\frac{8}{100}\cdot x \\ y(x)=15000+0.08\cdot x \end{gathered}$

The function that represents his yearly salary is "y(x)=15000+0.08*x".

To find his salary when he sells $500,000 in a year we need to make x equal to that value. Therefore:

$\begin{gathered} y(500000)=15000+0.08\cdot500000 \\ y(500000)=15000+40000 \\ y(500000)=55000 \end{gathered}$

His salary will be $55,000 for that year.

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The function g(x)= 112 Ln (0.121x) + 2011 models the year in which the population of New York City will equal x million people.

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We are given function: $g(x)= 112\ ln (0.121x) + 2011$ .

Given function models a particular year of population of New York City.

x represents population of New York City ( In millions).

We need to estimate the population of New York City in 2020.

Because g(x) function represents a particular year of population of New York City and we are given year 2020, so we need to replace g(x) by 2020 and solve for x.

Replacing g(x) by 2020, we get

$2020= 112\ ln (0.121x) + 2011$ : <em>This is the required equation to estimate the population of New York City in 2020</em>

Let us solve the above equation for x now.

$2020=112\ln \left(0.121x\right)+2011$

$\mathrm{Switch\:sides}$

$112\ln \left(0.121x\right)+2011=2020$

$\mathrm{Subtract\:}2011\mathrm{\:from\:both\:sides}$

$112\ln \left(0.121x\right)+2011-2011=2020-2011$

$Simplify$

$112\ln \left(0.121x\right)=9$

$\mathrm{Divide\:both\:sides\:by\:}112$

$\frac{112\ln \left(0.121x\right)}{112}=\frac{9}{112}$

$\mathrm{Simplify}$

$\ln \left(0.121x\right)=\frac{9}{112}$

$\mathrm{Apply\:log\:rule}:\quad \:a=\log _b\left(b^a\right)$

$\frac{9}{112}=\ln \left(e^{\frac{9}{112}}\right)$

$\ln \left(0.121x\right)=\ln \left(e^{\frac{9}{112}}\right)$

$0.121x=e^{\frac{9}{112}}$

$\mathrm{Divide\:both\:sides\:by\:}0.121$

$\frac{0.121x}{0.121}=\frac{e^{\frac{9}{112}} }{0.121}$

$x=8.95598$ ≈ 9 million people.

So, we can say..

Population of New York City in 2020 would be 8.95598 millions or 9 million ( approximately).

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3 years ago