Television
Jim’s mother watches a lot of TV. The screen is 30 inches tall and the bottom of the screen is 6 inches above her eye level while sitting in her favorite chair. To optimize her viewing experience, Jim wishes to calculate how far from the screen she should sit. Find the horizontal distance from the TV’s wall to her eyes that maximizes the angle her eyes trace from the bottom to the top of the screen.
1.draw pictures of at least THREE different cases (including the endpoints if any).
2. define all variables CLEARLY (with words) and then use those variables consistently.
3. Employ calculus to find the OPTIMAL case (which will be either a critical point or an endpoint).
To solve this problem, we can follow the steps outlined below:
1. Draw pictures of at least three different cases (including the endpoints if any):
Case 1: The TV is placed very close to the wall, and Jim's mother sits close to the TV.
Case 2: The TV is placed slightly further away from the wall, and Jim's mother sits at an intermediate distance.
Case 3: The TV is placed far from the wall, and Jim's mother sits at a large distance.
2. Define all variables clearly:
Let "D" represent the horizontal distance from the TV's wall to Jim's mother's eyes (in inches).
Let "H" represent the height of the TV screen (in inches).
Let "E" represent the height of Jim's mother's eyes from the floor (in inches).
Let "L" represent the height of Jim's mother's viewing line (distance from the floor to her eyes when seated, in inches).
3. Employ calculus to find the optimal case:
The objective is to maximize the angle that Jim's mother's eyes trace from the bottom to the top of the TV screen. To achieve this, we need to maximize the value of the angle tangent (θ) at the point where her eyes meet the TV screen.
The tangent of θ is given by tan(θ) = (H - E) / D. Rearranging this equation, we get D = (H - E) / tan(θ).
We need to find the value of D for which the angle θ is maximized. To do this, we need to consider the endpoints and critical points.
Endpoints:
- When D approaches zero, Jim's mother would be right in front of the TV screen. In this case, the angle θ would be 90 degrees, which is the maximum angle.
- When D approaches infinity, Jim's mother would be extremely far from the TV screen. In this case, the angle θ would approach 0, as the screen would appear almost flat.
Critical point:
- To find the critical points, we can take the derivative of D with respect to θ and set it equal to zero. Then solve for θ.
With this information, we can determine the optimal case by considering the endpoints and critical points of the function.