Day 2 — Quadratic Equations
Time: ~30 min · Topic: quadratic equations
The problem
A ball is thrown upward from the top of a 48-foot building at 32 ft/s. Its height after t seconds is h = -16t^2 + 32t + 48. When does the ball hit the ground?
Hint: What is the height when the ball reaches the ground? Set up that equation, then look for a common factor to simplify before solving.
Try it yourself
Take a few minutes. Don't peek at the steps. If you get stuck, the hint above is enough to nudge you forward.
Step-by-step solution
- Set h = 0 (ground level): -16t^2 + 32t + 48 = 0
- Divide every term by -16 to simplify: t^2 - 2t - 3 = 0
- Factor the quadratic: (t - 3)(t + 1) = 0, so t = 3 or t = -1
- Time can't be negative, so we discard t = -1. The ball hits the ground at t = 3 seconds.
Answer: The ball hits the ground after 3 seconds.
Why this matters
Every time you toss your phone onto your bed, its path through the air follows a perfect parabola — the same curve described by a quadratic equation.
This day in math: The Mathematician Who Won Olympic Silver
April 22, 1887 · Harald Bohr
On this day in 1887, Harald Bohr was born in Copenhagen, Denmark. Before becoming one of the 20th century's most original mathematicians, Bohr was a star footballer who won a silver medal at the 1908 London Olympics. Playing for Denmark, he helped crush France 17-1 in the semi-final — still an Olympic record. When he later defended his doctoral thesis in mathematics, the audience reportedly contained more football fans than mathematicians.
Why it matters: Bohr went on to create the theory of almost periodic functions, a powerful generalization of periodic behavior that influences signal processing, differential equations, and number theory to this day. He was also the brother of Nobel Prize-winning physicist Niels Bohr — making them perhaps the most brilliant sibling duo in scientific history.
Did you know?
There's a shape called a Reuleaux triangle that has three curved sides and three pointy corners — yet it rolls as smoothly as a perfect circle. Even wilder: engineers use drill bits shaped like Reuleaux triangles to cut holes that are nearly perfect squares!
A Reuleaux triangle is constructed by drawing three circular arcs, each centered on one corner of an equilateral triangle and passing through the other two corners. This gives it a property called "constant width" — the distance between any two parallel lines squeezing the shape is always the same, just like a circle. When it rotates inside a square boundary, its corners trace a path that covers about 98.8% of the square's area, producing a nearly square hole.
Done?
- Solved the problem (or read through the steps)
- Read the history note
- Read the "did you know" fact