Solving Quadratic Equations To Determine Mask Sales The First Day At Least 100 Masks Were Sold

Introduction

In the realm of mathematical problem-solving, quadratic equations often serve as powerful tools for modeling real-world phenomena. Consider the equation $m = d^2 - 12d + 45$, a mathematical expression that elegantly captures the relationship between the number of masks sold during a school fundraiser and the corresponding day of the fundraiser. This equation, a quadratic masterpiece, allows us to delve into the intricacies of mask sales, predicting the number of masks sold on any given day within the first three weeks of the fundraiser.

Our mission is to unravel the mystery of when the school first achieved the remarkable feat of selling at least 100 masks. To embark on this mathematical journey, we must harness the power of the quadratic equation, transforming it into an inequality that encapsulates our goal. We seek to determine the day, denoted by the variable 'd', when the number of masks sold, represented by 'm', reaches or surpasses the coveted 100-mask milestone. This transformation sets the stage for a fascinating exploration of quadratic inequalities and their solutions.

Setting up the Inequality

To uncover the pivotal day when at least 100 masks were sold, we must first translate the problem's narrative into a precise mathematical statement. We seek the day 'd' when the number of masks sold, 'm', equals or exceeds 100. This translates to the inequality: $m ≥ 100$. However, we have a powerful equation that links 'm' and 'd': $m = d^2 - 12d + 45$. By substituting this equation into our inequality, we forge a direct connection between 'd' and the target of 100 masks. This substitution yields the quadratic inequality: $d^2 - 12d + 45 ≥ 100$.

Now, we embark on a journey to simplify this inequality, transforming it into a more manageable form for analysis. Subtracting 100 from both sides of the inequality, we arrive at the equivalent inequality: $d^2 - 12d - 55 ≥ 0$. This seemingly simple step unveils the true nature of our problem – a quadratic inequality that holds the key to unlocking the day when mask sales soared to new heights. The stage is set for the next act, where we confront this quadratic inequality head-on, seeking its solutions and the insights they hold.

Solving the Quadratic Inequality

To conquer the quadratic inequality $d^2 - 12d - 55 ≥ 0$, we first embark on a quest to find the roots of the corresponding quadratic equation: $d^2 - 12d - 55 = 0$. These roots, the points where the quadratic expression equals zero, serve as crucial signposts along our path to understanding the inequality. We can employ the versatile quadratic formula to unearth these roots. The quadratic formula, a cornerstone of algebra, states that for an equation of the form $ax^2 + bx + c = 0$, the roots are given by: $x = (-b ± √(b^2 - 4ac)) / (2a)$.

In our case, we have $a = 1$, $b = -12$, and $c = -55$. Plugging these values into the quadratic formula, we embark on a calculation that will reveal the roots of our equation. After careful calculation, we arrive at the roots: $d = (12 ± √(12^2 - 4 * 1 * (-55))) / (2 * 1)$. Simplifying further, we get: $d = (12 ± √(144 + 220)) / 2$, which leads to: $d = (12 ± √364) / 2$. Approximating the square root of 364 as approximately 19.08, we find the roots to be approximately $d ≈ (12 + 19.08) / 2 ≈ 15.54$ and $d ≈ (12 - 19.08) / 2 ≈ -3.54$. These roots, one positive and one negative, mark the points where our quadratic expression crosses the x-axis.

With the roots in hand, we construct a sign chart, a visual aid that helps us discern the intervals where the quadratic expression is positive or negative. The sign chart is divided into intervals based on the roots, and we test a value within each interval to determine the sign of the expression. For our inequality, the critical points are approximately -3.54 and 15.54. Since we are dealing with days of a fundraiser, we only consider the positive root and values of 'd' that are practically meaningful (i.e., positive integers). Thus, we focus on the interval $d ≥ 15.54$.

Determining the First Day

Having navigated the complexities of the quadratic inequality, we now stand on the verge of unveiling the answer. The inequality $d^2 - 12d - 55 ≥ 0$ holds true when $d ≤ -3.54$ or $d ≥ 15.54$. However, in the context of our problem, the negative solution is meaningless, as we cannot have a negative day of a fundraiser. Therefore, we focus on the condition $d ≥ 15.54$.

Recall that 'd' represents the day of the fundraiser. Since days are whole numbers, we seek the smallest whole number that satisfies the condition $d ≥ 15.54$. This number, the first day when at least 100 masks were sold, is 16. Thus, the school achieved its goal of selling 100 masks or more on the 16th day of the fundraiser.

Conclusion

In this exploration of quadratic equations and inequalities, we successfully determined the day when a school fundraiser first sold at least 100 masks. We started with the equation $m = d^2 - 12d + 45$, a mathematical model that captured the relationship between mask sales and the day of the fundraiser. By translating the problem into an inequality, $d^2 - 12d - 55 ≥ 0$, we set the stage for a mathematical journey.

We then embarked on a quest to solve this quadratic inequality, employing the quadratic formula to find the roots of the corresponding equation. These roots, the signposts along our path, guided us in constructing a sign chart, a visual aid that illuminated the intervals where the inequality held true. Finally, by considering the practical constraints of our problem, we arrived at the answer: the school first sold at least 100 masks on the 16th day of the fundraiser.

This journey showcases the power of mathematics to model and solve real-world problems. Quadratic equations, with their elegant curves and intriguing properties, provide a lens through which we can understand and predict the behavior of phenomena as diverse as mask sales and projectile motion. As we conclude this exploration, we carry with us a deeper appreciation for the beauty and utility of mathematics in unraveling the mysteries of our world.

Therefore, the first day that at least 100 masks were sold is the 16th day.