a fence 20 feet tall runs parallel to a tall building at a distance of 5 feet from the building as shown in the diagram.
find the length of the shortest ladder that will reach from the ground over the fence to the wall of the building.
a. find a formula for the length of the ladder in terms of $\theta$. hint: split the ladder into 2 parts.

$l(\theta) =$

b. find the first derivative, $l(\theta)$.

$l(\theta) =$

c. find the critical number(s) of $l(\theta)$. leave in exact form.

$\theta =$

d. after verifying that the critical number optimizes the length of the ladder, calculate the length of the shortest ladder. round the results to three decimal places.

$l(\theta_{\min}) \approx$ feet

Question

a fence 20 feet tall runs parallel to a tall building at a distance of 5 feet from the building as shown in the diagram.
find the length of the shortest ladder that will reach from the ground over the fence to the wall of the building.
a. find a formula for the length of the ladder in terms of $\theta$. hint: split the ladder into 2 parts.

$l(\theta) =$

b. find the first derivative, $l(\theta)$.

$l(\theta) =$

c. find the critical number(s) of $l(\theta)$. leave in exact form.

$\theta =$

d. after verifying that the critical number optimizes the length of the ladder, calculate the length of the shortest ladder. round the results to three decimal places.

$l(\theta_{\min}) \approx$ feet

Accepted Answer

{ "quality": "clear", "question_count": 4, "skills_matched": [ "step_cot", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [], "new_concepts": [ "Trigonometric Optimization", "Ladder Over Fence Problem" ], "current_concepts": [ "Trigonometric Optimization", "Ladder Over Fence Problem", "Critical Points", "Derivative Equations" ] } ## Express the ladder length $L(\theta)$ $$ \begin{aligned} L_1 &= \frac{20}{\sin\theta} = 20\csc\theta \ L_2 &= \frac{5}{\cos\theta} = 5\sec\theta \ L(\theta) &= 20\csc\theta + 5\sec\theta \end{aligned} $$ ## Compute the first derivative $L'(\theta)$ $$ \begin{aligned} L'(\theta) &= \frac{d}{d\theta}(20\csc\theta + 5\sec\theta) \ L'(\theta) &= -20\csc\theta\cot\theta + 5\sec\theta\tan\theta \end{aligned} $$ ## Find the critical number $\theta$ $$ \begin{aligned} -20\csc\theta\cot\theta + 5\sec\theta\tan\theta &= 0 \ \frac{5\sin\theta}{\cos^2\theta} &= \frac{20\cos\theta}{\sin^2\theta} \ \tan^3\theta &= 4 \ \theta &= \arctan(\sqrt[3]{4}) \end{aligned} $$ ## Calculate the minimum length $L(\theta_{\min})$ $$ \begin{aligned} \theta_{\min} &= \arctan(4^{1/3}) \approx 1.0084 \text{ rad} \ L(\theta_{\min}) &= 20\csc(1.0084) + 5\sec(1.0084) \ L(\theta_{\min}) &= (20^{2/3} + 5^{2/3})^{3/2} \approx 33.004 \text{ feet} \end{aligned} $$ ## Question A A. Find a formula for the length of the ladder in terms of $\theta$. Hint: Split the ladder into 2 parts. $L(\theta) =$ $20\csc\theta + 5\sec\theta$ ## Question B B. Find the first derivative, $L'(\theta)$. $L'(\theta) =$ $-20\csc\theta\cot\theta + 5\sec\theta\tan\theta$ ## Question C C. Find the critical number(s) of $L(\theta)$. Leave in exact form. $\theta =$ $\arctan(\sqrt[3]{4})$ ## Question D D. After verifying that the critical number optimizes the length of the ladder, calculate the length of the shortest ladder. Round the results to three decimal places. $L(\theta_{\min}) \approx$ $33.004$ feet { "subject": "Mathematics", "question_type": "Multi-part", "knowledge_point": [ "Mathematics", "Calculus", "Optimization Problems" ] }

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QUESTION IMAGE

Question

Explanation:

Express the ladder length \(L(\theta)\)

Compute the first derivative \(L'(\theta)\)

Find the critical number \(\theta\)

Calculate the minimum length \(L(\theta_{\min})\)

Answer:

Question A

Question B

Question C

Question D