### Optimization Problem**Problem Statement:**5. *Find the area of the largest rectangle that can be inscribed in a right triangle with legs of lengths 4 cm and 5 cm if two sides of the rectangle lie along the legs.***Solution Steps:**1. **Define the Variables:** - Let the lengths of the legs of the right triangle be \(a = 4 \text{ cm}\) and \(b = 5 \text{ cm}\). - Let the dimensions of the rectangle be \(x \text{ cm}\) and \(y \text{ cm}\).2. **Determine the Relationship between \(x\) and \(y\):** - Because the right triangle has legs of lengths 4 cm and 5 cm, the relationship between \(x\) and \(y\) is given by the equation \( \frac{x}{a} + \frac{y}{b} = 1 \).3. **Formulate the Area Function to be Maximized:** - The area \(A\) of the rectangle is given by \(A = xy\).4. **Substitute \(y\) in terms of \(x\) into the Area Function:** - Using the relationship \( \frac{x}{4} + \frac{y}{5} = 1 \) to express \(y\) in terms of \(x\), we get: \[ y = 5 \left(1 - \frac{x}{4}\right) = 5 - \frac{5x}{4}. \] - Substitute \(y\) into the area function: \[ A = x \left(5 - \frac{5x}{4}\right) = 5x - \frac{5x^2}{4}. \]5. **Differentiate the Area Function to Find Maximum Area:** - Differentiate \(A\) with respect to \(x\): \[ \frac{dA}{dx} = 5 - \frac{5x}{2}. \] - Set the derivative equal to zero to find the critical points: \[ 5 - \frac{5x}{2} = 0 \implies x = 2. \]6. **Determine the Corresponding \(y\) Value:** - Substitute \(x

Answered: Image /qna-images/answer/fc8914cf-301e-473b-a7fa-106a30775f35.jpg

Answered: Optimizartion 5) Find the avea of the…

Optimizartion 5) Find the avea of the largest rectangle an be inscribed in a right triangle with lee lenytns 4cm and 5 cm' it two sides of th lie along the legs

Calculus: Early Transcendentals

8th Edition

ISBN:9781285741550

Author:James Stewart

Publisher:James Stewart

Chapter1: Functions And Models

Section: Chapter Questions

Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...

See similar textbooks

Similar questions

Find all right triangles sides of length 7 and 5, rounding answers to 3 decimal places, and give the area of each triangle.
Complete the dimensions of the given triangle
8. To temporarily support a wall during construction, the workers allow the wall to tilt towards them at an angle of 85° to the floor. If they support the wall with a piece of wood 8 ft up the wall and nailed 6 ft from the base of the wall, how long does this support need to be?
Please Anwser these question 4-8 and Make easy to read
Helpe me trigonometry please it's due tomorrow
YCCTK60H9_eFSwfSgso_emCn ALICIA CHEESE-9.. S Schoology ntro to Right Triangles 22 - 5609199.2 A Add-ons Help Last edit was made 2 minutes ago by M Arial BIUA 11 +. 2 手手主主王 Solve for X in each right triangle: 8. 1) 6.
2) BC = (3x + 25) B"C" = (5x- 5) Find the length of B'C: H. B' A' A" O a) 15 O b) 30 c) 50 O d) 70 Cannot be determined from the information given
3) HG) 5725+1 l00 t 106 Plot the polos & zoros
1) A rectangular garden has a length of 3 + 7 feet and a width of 2 + V7 feet. a. Find the area of the rectangular garden. b. The shaded region has an area of 2 - 3v7 square feet. Find the area of the unshaded region so that a flower bed can be put in. A = LW
80) solve the triangle a=7, b=1, C=170'
Triangle ABC has sides AB = 20 m, BC = 40 m, and CA = 50 m. if point D`is on the midpoint of line BC, determine the locatiori of polnt E from the verfex A along line AB such that the line DE willi divide the lot having equal An observer at point A observed the angle of. elevation of 27 to the top of the pole. If he move horizontally a distance "S" towards B, nearer to the pole, the angle of elevation of the pole at B is double that of A. Find the height of the pole. a) S Sin 48' b) S Sin 27 2. areas., a) 0 b) 1.4 c) 2.5 d) 3.7 c). S Sin 54. d) S Sin 45 1. Points A and B are 100 m. apart horizontally and are of the same elevation as the foot of a building. The angles of elevation of the top of the building from A and B are 21" and 32 respectively. How far is A fro the building? a), 259.28 m b) 235.56 m 4. Towers A and B stand on a leyel ground. From the top of the tower A which is 30 m. high the angle, of elevation of the top of tower B is 48". From the same point, the angle of…