Question

This is Calculus 1 Problem! MUST SHOW ALL THE JUSTIFICATION!!!

Answer 1

Given: A surveyor standing 50 feet from the base of a large tree measures the angle of elevation to the top of the tree as 75.8 degrees.

Required: To determine how accurately the angle must be measured if the percent error in estimating the tree's height is less than 5%.

Explanation: To estimate the angle, we will use the trigonometric ratio

`tanx=(h)/(50)\text{ ...\lparen1\rparen}`

where h is the tree's height, and x is the angle of elevation to the top of the tree.

Hence we get

`\begin{gathered} h=50\cdot(tan75.8\degree) \\ h=197.59\text{ feet} \end{gathered}`

Now differentiating equation 1, we get

`sec^2xdx=(1)/(50)dh`

We can write the above equation as:

`sec^2x\cdot(xdx)/(x)=(h)/(50)\cdot(dh)/(h)\text{ ...\lparen2\rparen}`

Also, it is given that the error in estimating the tree's height is less than 5%.

So

`(dh)/(h)=0.05`

Also, we need to convert the angle x in radians:

`x=1.32296\text{ rad}`

Putting these values in equation (2) gives:

`(dx)/(x)=(197.59)/(50)\cdot(cos^2(1.32296))/(1.32296)\cdot0.05`

Solving the above equation gives:

`\begin{gathered} (dx)/(x)=3.9518\cdot0.04548551012\cdot0.05 \\ =0.008987\text{ radians} \end{gathered}`

Let

`d\theta\text{ be the error in estimating the angle.}`

Then,

`\lvert{d\theta}\rvert\leq0.008987\text{ radians}`

Final Answer:

`\lvert{d\theta}\rvert\leq0.008987\text{ radians}`