Sara and Johnny visit grandpa Barney
Johnny loves his grandpa Barney. He is his mom’s dad. At one time, he used to visit him every summer but now, he has not gone to his place for almost five years. I guess his mom got too busy with her job and the house work. He asked her if he could borrow her car to visit grandpa Barney. Grandpa Barney lived in another town. Johnny thought that he could go there and come back the same day. Mom said yes. Johnny asked his girlfriend Sara if she wanted to come along. Sara was always looking to be with her sweetheart and readily agreed.
It was a nice summer day and the love birds enjoyed the two hour drive to grandpa Barney’s house. Grandpa was happy to see Johnny.
Johnny: Grandpa, on the phone I told you about my girlfriend Sara. This is her.
Grandpa: Good seeing you Sara. Johnny talks about you all the time.
Sara: Glad to see you grandpa. You seem to be in pain, why ?
Stairs are too steep
Grandpa: Yes, it is the arthritis in my knees. It hurts. You know, what is worse ? I have not gone outside the house for almost a month. I am locked in here.
Johnny: Why grandpa ?
Grandpa: Didn’t you see the stairs ? They are so steep. I fear that I could fall on my way out. Coming back would be equally bad. Even with my cane, it is very difficult to climb them.
Johnny: Sara, what can we do to help grandpa ?
Sara: That would be another Trig lesson for you. How do you want to start ?
Johnny: I want to start by measuring the rise (height) and run of the stairs.
Johnny got a measuring tape and then said: Sara, the total rise of the 4 stairs is 1.2 meters and the run is 1 meter.
Sara drew a picture and then said: That’s steep. The rise (length of BC) over run (the length of AB) is 1.2. That means tan of the angle BAC is 1.2. My calculator shows that arctan (1.2) is 50⁰. I have never seen that much rise over run in any other stairs. No wonder your knees hurt climbing them, grandpa Barney.
Jonny: Not just that, the rise of each stair is one fourth of 1.2 meters which is 30 centimeters. It is hard for him to lift his foot that much. So what should we do ?
Climb the same height in 8 steps rather than 4
Sara: Why don’t we make him climb the same height in 8 steps rather than 4. Then the rise of each step will be only 15 centimeters instead of 30.
Johnny: If we keep the run to be the same as before which is 25 centimeters per stair, for all the stairs we would need a total run of 8 x 25 centimeters which is 2 meters. I guess there is plenty of space in the front to accommodate that.
Sara: Now, the rise/run will be 1.2/2 = 0.6. That means tan (BAC) = 0.6. Arctan (0.6) = 31⁰.
Johnny: So climbing at an angle of 31⁰ will be much easier than at the 50⁰ angle. Grandpa, you know my dad is in the construction business. I will ask him to send someone to get you new stairs which are easier to climb.
Sara: Ask your dad to put railings on both the sides – left and the right. That will make it safer and easier.
Johnny and Sara had lunch with grandpa and then drove back. Johnny told mom and dad what had happened. Dad agreed to send someone to make the stairs. Mom was happy for two reasons. First, that it would help her father and second that Johnny had chosen a smart girlfriend, but she already knew that.
Time will come when grandpa will get older and he will need to go in and out of the house using a wheel chair. The ramp for the wheel chair should be no steeper than 6⁰. He would have to climb the same rise of 1.2 meters. What will be the run of this ramp ?
Solution: Make a triangle ABC with base AB and height BC.
BC = 1.2 meters
Angle BAC = 6⁰.
BC/AB = tan 6⁰ = 0.105.
Therefore, AB = BC/0.105 = 1.2/0.105 = 11.4 meters.
We don’t know if there is that much room in the yard of his house.
What solutions can you think of ? Use your imagination and discuss with others.