Word Problem Involving Optimizing Area By Using A Quadratic Function
Meet Barnaby Buttercup, a man whose life ambition was to create the most magnificent pumpkin patch the world had ever seen. Barnaby wasn’t just any gardener; he was a pumpkin whisperer, a squash sculptor, and a gourdcaster of dreams. His current dilemma, however, was not about the perfect shade of orange, but about the perfect fence.
Barnaby had a grand vision for his prize-winning pumpkins, each one destined for glory at the annual County Fair. But to protect his precious gourds from mischievous squirrels and overly enthusiastic toddlers, he needed a sturdy fence. He had a finite amount of fencing material, a magical roll that seemed to stretch and shrink based on Barnaby’s mood.
This year, Barnaby had a generous 400 feet of fencing to play with. He wanted to build a rectangular enclosure, a safe haven for his pumpkins to grow plump and proud. The bigger the area, the more pumpkins he could fit, and the more pumpkins, the higher his chances of winning the coveted "Golden Gourd Award."
He paced back and forth, his brow furrowed in concentration, muttering to himself. "If I make it long, it will be narrow. If I make it wide, it will be short. How can I get the most oomph out of my fence?" He imagined his pumpkins, yearning for space to bask in the sun and stretch their vines.
Barnaby wasn't a mathematician by trade. His expertise lay in compost and pollination, not calculus and parabolas. Yet, he felt a pull, a certain je ne sais quoi, to find the absolute best shape for his enclosure. He didn't want a "good enough" fence; he wanted the "Spectacularly Spacious Squash Sanctuary."
He decided to experiment. He imagined a fence 100 feet long. This would leave him with 200 feet for the other three sides (since one side would be, say, the side of his barn, a clever trick he learned from his Aunt Mildred). So, he could make two sides of 100 feet each, leaving 200 feet to be split between the remaining two sides. That meant each would be 100 feet. A perfect square! The area would be 100 x 100 = 10,000 square feet. "Not bad," he mumbled, picturing rows and rows of potential champions.
But then he thought, "What if I get a bit more adventurous?" He decided to try a length of 150 feet. With 400 feet of fencing, that meant 150 + 150 = 300 feet used for the two long sides. That left him with 100 feet for the two shorter sides. So, each shorter side would be 50 feet. The area in this case would be 150 x 50 = 7,500 square feet. "Hmm, that's less area," he grumbled, his brow furrowing deeper. His pumpkins seemed to droop in his imagination.
This was getting confusing. Barnaby needed a system. He pulled out a dusty notebook, once used for recording his prize-winning fertilizer recipes, and started scribbling. He let the length of one side be 'L' and the width of the other side be 'W'. The total fencing used would be 2L + 2W = 400. He then wanted to maximize the area, which is Area = L x W.
He bravely tackled the equation. From 2L + 2W = 400, he could simplify it to L + W = 200. This meant that the width (W) was equal to 200 - L. He substituted this into the area equation: Area = L x (200 - L). This looked like Area = 200L - L².
Now, this equation, Area = 200L - L², was a bit of a puzzle. Barnaby, with his limited math knowledge, saw the L² term and remembered something from his school days about things going "up and then down." He pictured a little hill, or perhaps a charmingly lopsided pumpkin. He knew that if he made the length very small, say 1 foot, the width would be almost 200 feet, and the area would be tiny. Similarly, if he made the length almost 200 feet, the width would be very small, and the area would again be tiny.
He thought about where the peak of that little pumpkin-shaped hill might be. He vaguely recalled something about the middle. If the total fencing (L + W) was 200, and he wanted L and W to be as close as possible to each other to maximize the area, then they should be equal. So, L = W. If L + W = 200 and L = W, then 2L = 200, which means L = 100. And if L = 100, then W = 100 too!
A square! A perfect, glorious square! Barnaby let out a triumphant "Huzzah!" His heart swelled with the sheer elegance of it all. He had found the mathematical secret to the most spacious pumpkin patch. The area would be 100 feet x 100 feet, giving him a magnificent 10,000 square feet of pumpkin paradise.
He envisioned his pumpkins, luxuriating in their perfectly proportioned domain. No more cramped quarters, no more squabbles over sunbeams. They would be plump, happy, and ready to woo the judges at the County Fair. This wasn't just about fencing; it was about creating an environment where dreams could grow, quite literally.
Barnaby realized that this little bit of math, this "quadratic function" as he'd later learn it was called, was like a secret ingredient. It wasn't just about finding the biggest area; it was about finding the smartest way to use his precious fencing. It was about understanding the beautiful dance between length and width.
He imagined other people facing similar dilemmas. A baker wanting the largest cake for a birthday, a farmer wanting the most grazing land for their sheep, or even a child wanting the biggest blanket fort. The same principle, the magic of finding that perfect balance, could apply. It was a delightful, heartwarming thought.
He chuckled, picturing his neighbour, Agnes Periwinkle, who always insisted on long, narrow gardens. "Poor Agnes," he mused, "her tomatoes are probably feeling quite cooped up." He decided he might share his newfound knowledge, perhaps over a cup of tea and a slice of her famously lopsided zucchini bread.
The next day, Barnaby went to work with renewed vigor. He carefully measured and staked out his 100x100 foot square. The fencing unrolled smoothly, almost as if it knew it was being used for a purpose of such profound, pumpkin-loving importance. He hummed a happy tune, a melody of geometry and gourds.
As he worked, he imagined the future. He saw himself, crowned with the Golden Gourd Award, his pumpkins glowing with pride, and all thanks to a simple equation that helped him find the optimal area. It was a sweet victory, a testament to the fact that sometimes, even the most down-to-earth pursuits can lead to some surprisingly beautiful mathematical discoveries. And that, Barnaby thought with a contented sigh, was truly something to be celebrated.