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There are many problems that deal with shadows. Usually these problems use the relationships found in trigonometry or similar triangles. I have listed below three of my favorite shadow problems,
1) If I am 6 feet tall (in my dreams) and I cast a 5 foot shadow, how tall is a flagpole that casts a 10 foot shadow? Answer: 12 feet
2) A street light is mounted at the top of a 15 foot pole. I am walking away from the pole at a rate of 5 ft/second. How fast is the tip of my shadow moving when I am 40 feet from the pole? Remember: I am 6 feet tall. :-) Answer: 25/3 ft/second
3) My favorite problem that applies the use of shadows is the calculation of the Earth's circumference by Eratosthenes over 2000 years ago. Eratosthenes estimate was 25,000 miles. The actual circumference is 24,902 miles.
Today, I complete my Sibling Bond series. My previous blogs were tributes to my sister and younger brothers. The day to day conflicts and agreements with these three siblings acted as a rock tumbler, shaping and polishing me in my progression to adulthood.
Today is my brother, Russ' birthday. Today, he is a decade older than I am. Unlike my other siblings, Russ was a comet, following an unpredictable, elliptical path that when his intersected mine, his effect on my growth was more transitory but not any less impactful.
Russ was sixteen when I was six. The memories of my family became more distinct for me at that age. My younger brothers were energetic, four year old playmates and my sister was the ever present baby sitter. My brother, however, had his driver license, was involved in sports in high school, and was working with my father in his shop. He flew in my view like a bird at my window feeder, who departs to locations unknown as soon as I am aware of its presence.
I treasured any time I had with him. I adored this mysterious, independent, good natured brother. He occasionally included me in his excursions. I was oblivious to the task at hand and eagerly followed behind at a pace that could quite never keep up to his confident gate. I recall many times that I was introduced to his friends, as "his little shadow", a description that caused me to grin and swell with pride.
When I was nine, Russ was a senior in high school and his orbit grew. I saw him less often but desired his attention even more. During this time, I was watching shows such as Dennis The Menace and the Little Rascals. Each of these shows had booby traps involved in their plots. I was captured with the idea of setting traps and Russ became my victim. For a period of three months, a series of pranks greeted him when he came home from his late night excursions. As he entered his room, a barrage of novelties fell from a precarious bucket perched at the top of the door. Books, shredded paper, and water were all items that landed on my brother's unsuspecting head. Throughout this period of time, Russ never raised his voice. He took it in stride. He was patient with his annoying brother.
I became a teenager in the early 1970's and Russ was in his early 20's. He invited me to stay with him in Minneapolis for a weekend. I recall walking through Dinky Town, encountering the sights, sounds, and smells that defined the 70's. Those sensory inputs opened the eyes of this naive, small town kid. That evening, we decided to go to a movie. Our choices were: Magnum Force, Paper Moon, The Paper Chase, and Papillon. We chose Papillon. The night ended, falling asleep while watching Rat Patrol, and I, dreaming of my urban life.
As I have grown older, Russ is no longer the super hero that I thought he was when I was younger. He, like us all, is flawed and complex, but he is wise. He is patient and calm. He has the gift of diminishing my failings and applauding my achievements. He is someone I aspire to be.
As I write this blog about my brother, I am surprised about the details I can recall about our times together. I feel his interactions with me were intentional. I believe he wanted to impact me differently than what I experiencing at home.
I have learned from him that family allows its members to find their own way and encourages them to stitch their own block on the family quilt. He has taught me that family welcomes all its members, and, if necessary, rejoices on the return of its members to the fold, and that as members of the family, we should refrain from casting judgement.
Happy Birthday Numero Unus!
3) My favorite problem that applies the use of shadows is the calculation of the Earth's circumference by Eratosthenes over 2000 years ago. Eratosthenes estimate was 25,000 miles. The actual circumference is 24,902 miles.
Today is my brother, Russ' birthday. Today, he is a decade older than I am. Unlike my other siblings, Russ was a comet, following an unpredictable, elliptical path that when his intersected mine, his effect on my growth was more transitory but not any less impactful.
Russ was sixteen when I was six. The memories of my family became more distinct for me at that age. My younger brothers were energetic, four year old playmates and my sister was the ever present baby sitter. My brother, however, had his driver license, was involved in sports in high school, and was working with my father in his shop. He flew in my view like a bird at my window feeder, who departs to locations unknown as soon as I am aware of its presence.
I treasured any time I had with him. I adored this mysterious, independent, good natured brother. He occasionally included me in his excursions. I was oblivious to the task at hand and eagerly followed behind at a pace that could quite never keep up to his confident gate. I recall many times that I was introduced to his friends, as "his little shadow", a description that caused me to grin and swell with pride.
When I was nine, Russ was a senior in high school and his orbit grew. I saw him less often but desired his attention even more. During this time, I was watching shows such as Dennis The Menace and the Little Rascals. Each of these shows had booby traps involved in their plots. I was captured with the idea of setting traps and Russ became my victim. For a period of three months, a series of pranks greeted him when he came home from his late night excursions. As he entered his room, a barrage of novelties fell from a precarious bucket perched at the top of the door. Books, shredded paper, and water were all items that landed on my brother's unsuspecting head. Throughout this period of time, Russ never raised his voice. He took it in stride. He was patient with his annoying brother.
I became a teenager in the early 1970's and Russ was in his early 20's. He invited me to stay with him in Minneapolis for a weekend. I recall walking through Dinky Town, encountering the sights, sounds, and smells that defined the 70's. Those sensory inputs opened the eyes of this naive, small town kid. That evening, we decided to go to a movie. Our choices were: Magnum Force, Paper Moon, The Paper Chase, and Papillon. We chose Papillon. The night ended, falling asleep while watching Rat Patrol, and I, dreaming of my urban life.
As I have grown older, Russ is no longer the super hero that I thought he was when I was younger. He, like us all, is flawed and complex, but he is wise. He is patient and calm. He has the gift of diminishing my failings and applauding my achievements. He is someone I aspire to be.
As I write this blog about my brother, I am surprised about the details I can recall about our times together. I feel his interactions with me were intentional. I believe he wanted to impact me differently than what I experiencing at home.
I have learned from him that family allows its members to find their own way and encourages them to stitch their own block on the family quilt. He has taught me that family welcomes all its members, and, if necessary, rejoices on the return of its members to the fold, and that as members of the family, we should refrain from casting judgement.
Happy Birthday Numero Unus!
