A picture is 5 feet high and the eye level of an observer is 2 feet below the bottom
edge of the picture. How far from the picture should the observer stand if he wants
to maximize the angle subtended by the picture? help me :((
draw the picture. let x be the distance to the wall.
draw let theta1 be the subtended angle of the picture, and theta2 be the angle subtended by thebottom of the picture and the horizontal.
In the lower triangle, the distance (hypotensue) from eye to bottom of pic is
sqrt (x^2+2^2)
The upper angle (top of pic to eye) is Theta3.
law of sines:
sinT3/x = sin90/sqrt(x^2+7^2)
solve for sineT3
in the upper triangle
sinT3/sqrt(2^2+x^2) = sinT1/5 an T1 is the subtended angle. Let y=sinT1, if you maximize T1, you maximize y
y=5/sqrt(4+x^2) *x/sqrt(49+x^2)
find dy/dx, set to zero, solve for x.
To maximize the angle subtended by the picture, the observer should stand at a distance where the picture is placed at eye level.
Let's break down the problem:
The height of the picture is 5 feet.
The eye level of the observer is 2 feet below the bottom edge of the picture.
To find the optimal distance, we can use basic trigonometry.
1. Calculate the total height from eye level to the top of the picture:
Total height = height of the picture + distance from eye level to the bottom edge
= 5 feet + 2 feet
= 7 feet
2. Now, draw a right-angled triangle with the vertical line representing the height of the picture and the horizontal line representing the distance from the observer to the picture. The hypotenuse will represent the line of sight from the observer to the top of the picture.
3. Let the distance between the observer and the picture be 'x' feet. We want to find the value of 'x' for which the height of the picture (7 feet) divides the hypotenuse into two equal parts.
4. Using the Pythagorean theorem, we know that the square of the hypotenuse equals the sum of the squares of the other two sides of a right-angled triangle. In this case, the equation becomes:
x^2 + 7^2 = (x + 5)^2
5. Simplify the equation:
x^2 + 49 = x^2 + 10x + 25
6. Subtract x^2 from both sides to eliminate the variable:
49 = 10x + 25
7. Simplify the equation further:
24 = 10x
8. Divide both sides by 10:
x = 24/10
x = 2.4
Therefore, the observer should stand approximately 2.4 feet away from the picture to have the maximum angle subtended by the picture.