i am considerably stuck, please help?

[rebeccaramirez]

hello this is a take home test question; nobody in my class can seem to figure the last part of this question out... i will type how far i have gotten, along with the problem


consider the curve -8x[sup:386dmdit]2[/sup:386dmdit] + 5xy + y[sup:386dmdit]3[/sup:386dmdit] = -149
a. find dx/dy
b. write an equation for the line tangent to the curve at the point (4, -1)
c. there is a number k so that the point (4.2, k) is on the curve. using the tangent line found in part b, approximate the value of k
d. write an equation that can be solved to find the actual value of k so that the point (4.2, k) is on the curve
e. solve the equation found in part d for the value of k


a. dy/dx = (16x - 5y)/(3y[sub:386dmdit]2[/sub:386dmdit] + 5x)
b. dy/dx (4, -1) = 3 which leads to the equation y+1=3(x-4)
c. (4.2, k) leads to k+1= 3x - 12 which leads to k= 12.6 - 13 which leads to k= -0.4
d. we came up with the equation k[sup:386dmdit]3[/sup:386dmdit] + 21k + 7.88 = 0
from here we are stuck... i was contemplating using Newtons method, but I fear it would be an entirely incorrect approach

if you can help me out, it would be very appreciated: i have been on break for three weeks now and my brain is quite dead, I cannot figure this out
thank you

 

[galactus]

Sure, Newton's method would be as good a choice as any. Give it a go.

You could factor your cubic, but that would be more complicated than Newton.

\(\displaystyle k^{3}+21k+\frac{197}{25}=0\)

factored:

\(\displaystyle (k+.372771439184)(k^{2}-.372771439184k+21.1389585459)=0\)

Now, it's easy to see the value of k, is it not?.

I would use Newton.

If you wanted to show off, you could use Cardano's formula for factoring cubics.

 

[fasteddie65]

If I remember correctly, this was on an AP Exam where calculators were allowed. Simply solve the cubic equation using either a graphical solution or the Solve function.