a billboard 20ft tall is located on top of a building with its lower edge 60 ft above the level of a viewers eye.
how far from a point directly below the sign should a viewer stand to maximize the angle (theta) between the lines of sight of the top and bottom of the billboard.
hint: let theta=a-b then tan (theta)=tan(a-b)=(tan(a)-tan(b))/(1+tan(a)tan(b)) differentiate implicitly with respect to x and set d(theta)/dx=0.
im going to try to draw the drawing but no promises on how it'll turn out.
theta=t
........a_
......./|20|
....../ |..|
...../t/|b.|
..../\/.|..|
..././..|.60
,,/\/...|..|
./\/\a..|..|
0./.\b\.|..|
|-----x-|c i hope that makes since!!(:
Using the hint,
tanθ = tan(a-b) = (tan a - tan b)/(1 + tana tanb)
If the viewer is at a distance x,
tan a = 80/x
tan b = 60/x
tanθ = (80/x - 60/x)/(1 + 80/x * 60/x)
tanθ = 20x/(x^2 + 4800)
sec^2θ θ' = 20(4800-x^2)/(4800 + x^2)^2
θ' = 20(4800-x^2)/((4800 + x^2)^2 * sec^2θ)
Now,
sec^2θ = 1 + tan^2θ = 1 + 40x^2/(4800+x^2)^2
so,
θ' = 20(4800-x^2)/(4800 + x^2)^2 * (4800+x^2)^2/(40x^2 + (4800+x^2)^2)
= 20(4800-x^2)/(40x^2 + (4800+x^2)^2)
θ' = 0 when 4800 = x^2 or,
x = 40√3 = 69.38'
Haha, don't worry! I'll do my best to understand your drawing. Let's solve this problem!
So, we're looking for the point (denoted by 'x') directly below the billboard from which a viewer should stand to maximize the angle (theta) between the lines of sight of the top and bottom of the billboard.
To solve this, let's consider the right triangle formed by the viewer's eye, the point directly below the billboard ('x'), and the bottom of the billboard. The height of the billboard is 20 ft (given), and the distance between the viewer's eye and 'x' is 'c'.
Now, to maximize the angle theta, we need to differentiate tan(theta) with respect to 'x' and set d(theta)/dx = 0, as hinted. Let's do that!
d(theta)/dx = (d/dx)[tan(a-b)]
Using the hint, we can apply the chain rule and simplify:
d(theta)/dx = [(1+tan(a)tan(b))(d/dx)(tan(a)-tan(b)) - (tan(a)-tan(b))(d/dx)(1+tan(a)tan(b))] / (1+tan(a)tan(b))^2
Now, let's find the values of tan(a) and tan(b) in terms of 'x', 'c', and the given dimensions of the triangle:
tan(a) = 60/('x'+c)
tan(b) = 60/('x')
Note: The height of the billboard (20 ft) does not affect the angles here, so we won't include it in our calculations.
Substituting the values of tan(a) and tan(b) into the differentiation expression, we get:
d(theta)/dx = [(1+(60/('x'+c))(60/'x'))((60/('x'+c))' - (60/'x')(d/dx)(1+(60/('x'+c))(60/'x'))] / (1+(60/('x'+c))(60/'x'))^2
To find the value of 'x' that maximizes theta, we need to solve the equation:
d(theta)/dx = 0
I hope this helps you in your calculations! Remember to simplify as much as possible and solve the equation for 'x' to find the optimal location for the viewer. Good luck!
Your drawing makes sense! Let's solve the problem step by step.
Step 1: Determine the relationship between the variables in the problem.
Let's consider the following variables:
- h: the height of the billboard (20 ft)
- d: the distance from the viewer to the point directly below the sign
- c: the distance from the viewer to the building
- a: the angle between the line of sight to the top of the billboard and the horizontal line
- b: the angle between the line of sight to the bottom of the billboard and the horizontal line
- θ: the angle between the lines of sight of the top and bottom of the billboard (θ = a - b)
Step 2: Relate the variables using trigonometry.
We can use the tangent function to relate the variables:
tan(a) = h / (d + c)
tan(b) = h / c
Step 3: Express θ in terms of a, b, and θ.
θ = a - b
Step 4: Differentiate implicitly with respect to x and set d(θ)/dx = 0.
To find the value of x that maximizes θ, we need to find the critical points where the derivative is equal to zero. We can differentiate θ implicitly with respect to x:
d(θ)/dx = d(a - b)/dx = d(a)/dx - d(b)/dx
Step 5: Express d(θ)/dx in terms of a, b, and their derivatives.
Using the chain rule, we can express d(a)/dx and d(b)/dx in terms of a, b, d, and c:
d(a)/dx = (dh/dx) / (d + c)
d(b)/dx = (dh/dx) / c
Step 6: Substitute the expressions for d(a)/dx and d(b)/dx into d(θ)/dx.
d(θ)/dx = (dh/dx) / (d + c) - (dh/dx) / c
d(θ)/dx = (dh/dx)(1/c - 1/(d+c))
Step 7: Set d(θ)/dx = 0 and solve for x.
To find the critical points, we set d(θ)/dx = 0:
(dh/dx)(1/c - 1/(d+c)) = 0
Step 8: Solve for x to determine the position of the viewer.
From step 7, we can see that either dh/dx = 0 or (1/c - 1/(d+c)) = 0.
- If dh/dx = 0, it means that the height of the billboard does not change with respect to x. In this case, there is no value of x that maximizes θ.
- If (1/c - 1/(d+c)) = 0, we can solve for x:
1/c = 1/(d+c)
1 = c / (d+c)
d+c = c
d = 0
Therefore, the viewer should stand directly below the sign (d = 0) to maximize the angle θ between the lines of sight of the top and bottom of the billboard.
Note: In this problem, we assumed that the viewer is looking straight ahead, and the line of sight of the viewer is parallel to the horizontal line.
Based on the diagram you provided, we can see that the viewer is standing at point x, directly below the sign. The height of the billboard (20 ft) is divided by two to get the individual heights of the top and bottom of the billboard, which we will call a and b, respectively. The lower edge of the billboard is 60 ft above the viewer's eye level.
To find the distance from the point directly below the sign where a viewer should stand to maximize the angle (theta) between the lines of sight of the top and bottom of the billboard, we need to apply the hint given.
Let's denote the distance from the viewer to the base of the building (point c) as d and the angle between the viewer's line of sight to the top of the billboard (a) and the horizontal line as angle a. Similarly, the angle between the viewer's line of sight to the bottom of the billboard (b) and the horizontal line will be angle b.
Since theta = a - b, we want to maximize theta by maximizing a and minimizing b. This can be done by maximizing the angle a with respect to d and minimizing the angle b with respect to d.
Now, let's differentiate the expression tan(theta) = tan(a-b) = (tan(a) - tan(b))/(1 + tan(a)*tan(b)) implicitly with respect to x (since d is dependent on x) and set d(theta)/dx = 0.
Differentiating the expression with respect to x:
(d(tan(theta))/dx) = (d(tan(a-b))/dx) = d/dx((tan(a) - tan(b))/(1 + tan(a)*tan(b)))
Apply the quotient rule of differentiation:
(d(tan(theta))/dx) = [(d(tan(a))/dx)*(1 + tan(a)*tan(b)) - (tan(a) - tan(b))*(d(1 + tan(a)*tan(b))/dx)] / (1 + tan(a)*tan(b))^2
Now, we want to find the critical point where d(theta)/dx = 0:
Set (d(tan(theta))/dx) = 0:
[(d(tan(a))/dx)*(1 + tan(a)*tan(b)) - (tan(a) - tan(b))*(d(1 + tan(a)*tan(b))/dx)] = 0
Simplifying the equation, we get:
(d(tan(a))/dx)*(1 + tan(a)*tan(b)) - (tan(a) - tan(b))*(d(1 + tan(a)*tan(b))/dx) = 0
Since we are trying to maximize angle a, we want tan(a) to be as large as possible. This occurs when a is a right angle (90 degrees), which gives us tan(a) = ∞. So, we have:
(∞/dx)*(1 + ∞*tan(b)) - (∞ - tan(b))*(d(1 + ∞*tan(b))/dx) = 0
Since the term (∞ - tan(b)) will be ∞, we know that (∞ - tan(b))*(d(1 + ∞*tan(b))/dx) will also be ∞.
Therefore, the equation becomes:
(∞/dx)*(1 + ∞*tan(b)) - ∞ = 0
Simplifying further, we have:
∞ - ∞ = 0
This equation indicates that there is no critical point where d(theta)/dx = 0. Hence, we cannot use this method to find the distance from the point directly below the sign where a viewer should stand to maximize the angle theta.
To find the answer, we may need to use an alternative method or seek additional information.