Mylo O'Moore

2021-08-19

To find:
The total sales T(x) of Northwest Creations, to the nearest dollar using the model where the distribution of x thousand catalogues is approximated by $T\left(x\right)=5000\mathrm{log}\left(x+1\right)$.

### Answer & Explanation

Fatema Sutton

Concept:
Modeling is a method of simulating the real life situations with mathematical equations. Using these models we can forecast the future behavior. We can translate the mathematical word problem into a math expression using variables.
Calculation:
The given model is $T\left(x\right)=5000\mathrm{log}\left(x+1\right)$
Where x is the number of catalogues distributed in throusands and T(x) is the total sales in dollars.
So, in order to get the total sales T(x), replace x in the given model with the number of catalogues.
a) $T\left(x\right)=5000\mathrm{log}\left(x+1\right)$
Replace x with $5000,T\left(5000\right)=5000\mathrm{log}\left(5000+1\right)$
Since $\mathrm{log}x={\mathrm{log}}_{10}x,T\left(5000\right)=5000{\mathrm{log}}_{10}5001$
Using calculator, $T\left(5000\right)\approx 5000×3.699$
$T\left(5000\right)\approx \mathrm{}18495$
Based on this model, the total sales of the Northwest Creations resulting from 5000 catalogues is $18495 b) $T\left(x\right)=5000\mathrm{log}\left(x+1\right)$ Replace x with $24000,T\left(24000\right)=5000\mathrm{log}\left(24000+1\right)$ Since $\mathrm{log}x={\mathrm{log}}_{10}x,T\left(24000\right)=5000{\mathrm{log}}_{10}24001$ Using calculator, $T\left(24000\right)\approx 5000×4.3802$ $T\left(24000\right)\approx \mathrm{}21901$ Based on this model, the total sales of the Northwest Creations resulting from the distribution of 24000 catalogues is$21901
c) $T\left(x\right)=5000\mathrm{log}\left(x+1\right)$
Replace x with $49000,T\left(49000\right)=5000\mathrm{log}\left(49000+1\right)$
Since $\mathrm{log}x={\mathrm{log}}_{10}x$, we have $T\left(49000\right)=5000{\mathrm{log}}_{10}49001$
Using calculator, $T\left(49000\right)\approx 5000×4.6902$
$T\left(49000\right)\approx \mathrm{}23451$
Based on this model, the total sales of the Northwest Creations resulting from 49000 catalogues is $23451 Final statement: Based on this model, the total sales of the Northwest Creations resulting from 5000 catalogues is$18495, from 24000 catalogues is $21901 and from 49000 catalogues is$23451.

Do you have a similar question?

Recalculate according to your conditions!