Answer:
c I am pretty sure unless my math is wrong
Step-by-step explanation:
Answer:
The rate of change of the distance
when x = 9 and y = 12 is
.
Step-by-step explanation:
This is an example of a related rate problem. A related rate problem is a problem in which we know one of the rates of change at a given instant
and we want to find the other rate
at that instant.
We know the rate of change of x-coordinate and y-coordinate:
![\frac{dx}{dt}=-2\:\frac{m}{s} \\\\\frac{dy}{dt}=-8\:\frac{m}{s}](https://tex.z-dn.net/?f=%5Cfrac%7Bdx%7D%7Bdt%7D%3D-2%5C%3A%5Cfrac%7Bm%7D%7Bs%7D%20%5C%5C%5C%5C%5Cfrac%7Bdy%7D%7Bdt%7D%3D-8%5C%3A%5Cfrac%7Bm%7D%7Bs%7D)
We want to find the rate of change of the distance
when x = 9 and y = 12.
The distance of a point (x, y) and the origin is calculated by:
![s=\sqrt{x^2+y^2}](https://tex.z-dn.net/?f=s%3D%5Csqrt%7Bx%5E2%2By%5E2%7D)
We need to use the concept of implicit differentiation, we differentiate each side of an equation with two variables by treating one of the variables as a function of the other.
If we apply implicit differentiation in the formula of the distance we get
![s=\sqrt{x^2+y^2}\\\\s^2=x^2+y^2\\\\2s\frac{ds}{dt}= 2x\frac{dx}{dt}+2y\frac{dy}{dt}\\\\\frac{ds}{dt}=\frac{1}{s}(x\frac{dx}{dt}+y\frac{dy}{dt})](https://tex.z-dn.net/?f=s%3D%5Csqrt%7Bx%5E2%2By%5E2%7D%5C%5C%5C%5Cs%5E2%3Dx%5E2%2By%5E2%5C%5C%5C%5C2s%5Cfrac%7Bds%7D%7Bdt%7D%3D%202x%5Cfrac%7Bdx%7D%7Bdt%7D%2B2y%5Cfrac%7Bdy%7D%7Bdt%7D%5C%5C%5C%5C%5Cfrac%7Bds%7D%7Bdt%7D%3D%5Cfrac%7B1%7D%7Bs%7D%28x%5Cfrac%7Bdx%7D%7Bdt%7D%2By%5Cfrac%7Bdy%7D%7Bdt%7D%29)
Substituting the values we know into the above formula
![s=\sqrt{9^2+12^2}=15](https://tex.z-dn.net/?f=s%3D%5Csqrt%7B9%5E2%2B12%5E2%7D%3D15)
![\frac{ds}{dt}=\frac{1}{15}(9(-2)+12(-8))\\\\\frac{ds}{dt}=\frac{1}{15}\left(-18-96\right)\\\\\frac{ds}{dt}=\frac{1}{15}(-114)=-\frac{38}{5}=-7.6\:\frac{m}{s}](https://tex.z-dn.net/?f=%5Cfrac%7Bds%7D%7Bdt%7D%3D%5Cfrac%7B1%7D%7B15%7D%289%28-2%29%2B12%28-8%29%29%5C%5C%5C%5C%5Cfrac%7Bds%7D%7Bdt%7D%3D%5Cfrac%7B1%7D%7B15%7D%5Cleft%28-18-96%5Cright%29%5C%5C%5C%5C%5Cfrac%7Bds%7D%7Bdt%7D%3D%5Cfrac%7B1%7D%7B15%7D%28-114%29%3D-%5Cfrac%7B38%7D%7B5%7D%3D-7.6%5C%3A%5Cfrac%7Bm%7D%7Bs%7D)
The rate of change of the distance
when x = 9 and y = 12 is ![-7.6\:\frac{m}{s}](https://tex.z-dn.net/?f=-7.6%5C%3A%5Cfrac%7Bm%7D%7Bs%7D)
Let j be Jacob's money. Then Angie is 3j, and Erin is 3j+35. So:
j+3j+3j+35=196
7j=161
j=23
Jacob had $23
☺☺☺☺