Relative Error of Right Triangle Area

[Jimbo57]

Homework Statement

Anyone mind checking my answer for me?

The area of a right triangle with hypotenuse H is

A=1/4H^2sin(2x)

x is one of the acute angles. Use differentials to estimate the relative errors of the area if H = 4cm exactly and x is measured to be 30degrees with an error of measurement of 15 minutes of arc.

Homework Equations

A=1/4H^2sin(2x)
dA=1/2H^2cos(2x)dx
dA/A=2cot(60)(0.25)

The Attempt at a Solution

A=1/4*16sin(60)=3.464cm^2
dA=8cos(60)(0.25)
=0.125cm^2
dA/A=0.125/3.464=0.2887=relative error

Now, did I have to convert to radians in step one? 30 to pi/6? Or is this right?

 

[Dick]

Homework Statement

Anyone mind checking my answer for me?

The area of a right triangle with hypotenuse H is

A=1/4H^2sin(2x)

x is one of the acute angles. Use differentials to estimate the relative errors of the area if H = 4cm exactly and x is measured to be 30degrees with an error of measurement of 15 minutes of arc.

Homework Equations

A=1/4H^2sin(2x)
dA=1/2H^2cos(2x)dx
dA/A=2cot(60)(0.25)

The Attempt at a Solution

A=1/4*16sin(60)=3.464cm^2
dA=8cos(60)(0.25)
=0.125cm^2
dA/A=0.125/3.464=0.2887=relative error

Now, did I have to convert to radians in step one? 30 to pi/6? Or is this right?

The formula d(sin(2x))=2*cos(2x)*dx is correct only if x is measured in radians. You can call the 2x part either 60 degrees or pi/3 if you adjust your calculator for degrees or radians, but the dx part definitely needs to be in radians.

 

[Jimbo57]

The formula d(sin(2x))=2*cos(2x)*dx is correct only if x is measured in radians. You can call the 2x part either 60 degrees or pi/3 if you adjust your calculator for degrees or radians, but the dx part definitely needs to be in radians.

Thanks Dick. So the steps I took were correct just needed to convert to radians?

How does this look?
15/60 degrees = 0.00436rad=dx
A=1/4*16sin(pi/3)=3.464cm^2
dA=8cos(pi/3)(0.00436)
=0.01744cm^2
dA/A=0.01744/3.464≈0.005=relative error

 

[Dick]

Thanks Dick. So the steps I took were correct just needed to convert to radians?

How does this look?
15/60 degrees = 0.00436rad=dx
A=1/4*16sin(pi/3)=3.464cm^2
dA=8cos(pi/3)(0.00436)
=0.01744cm^2
dA/A=0.01744/3.464≈0.005=relative error

That sounds much better.