mattgondek4

2021-08-14

To find:
The percent of moisture in the central Sierra Nevada (a mountain range in California), to the nearest tenth that falls as snow at each altitude (a) 3000 ft (b) 4000 ft (c) 7000 ft using the model $f\left(x\right)=86.3Inx-680$, where x is altitude in feet and f(x) is the percent of moisture that falls as snow.

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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 $f\left(x\right)=86.3Inx-680$
where x is altitude in feet and f(x) is the percent of moisture that falls as snow.
So, in order to get the percent of moisture f(x), replace x in the given model with the values of altitudes.
a) Let $x=3000ft$
$f\left(x\right)=86.3Inx-680$
Replace x with $3000,f\left(3000\right)=86.3\left(In3000\right)-680$
Using calculator, $f\left(3000\right)\approx 86.3×8.00636-680$
Multiplying, $f\left(3000\right)\approx 690.9495-680$
Subtracting, $f\left(3000\right)\approx 10.9495$
Rounded to nearest tenth, $f\left(3000\right)\approx 10.95$
Based on this model, $10.95\mathrm{%}$ of moisture falls as snow.
b) Lex $x=4000ft$
$f\left(x\right)=86.3Inx-680$
Replace x with $3000,f\left(4000\right)=86.3\left(In4000\right)-680$
Using calculator, $f\left(4000\right)\approx 86.3×8.294049-680$
Multiplying, $f\left(4000\right)\approx 717.776483-680$
Subtracting, $f\left(4000\right)\approx 35.7764$
Rounded to nearest tenth, $f\left(4000\right)\approx 35.78$
Based on this model, $35.78\mathrm{%}$ of moisture falls as snow.
c) Let $x=7000ft$
$f\left(x\right)=86.3Inx-680$
Replace x with $7000,f\left(7000\right)=86.3\left(In7000\right)-680$
Using calculator, $f\left(7000\right)\approx 86.3×8.853665-680$
Multiplying, $f\left(7000\right)\approx 764.071326-680$
Subtracting, $f\left(7000\right)\approx 84.07132$
Rounded to nearest tenth, $f\left(7000\right)\approx 84.07$
Based on this model, $84.07\mathrm{%}$ of moisture falls as snow.
Final statement:
Based on this model, $10.95\mathrm{%}$ of moisture falls as snow at 300 ft, $35.78\mathrm{%}$ of moisture falls as snow at 4000 ft and $84.07\mathrm{%}$ of moisture falls as snow at 7000 ft.

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