# Use the information below to determine which equation belongs to which sibling, as well as the object which was thrown and the age order.?

The DePeppo children, Ann, Debbie, Damian, and Adam created a contest to test their athletic abilities and physical strength. They threw objects into the air to see who was the best.

These four equations depict the travel of the objects they threw.

Use the information below to determine which equation belongs to which sibling, as well as the object which was thrown and the age order.

The youngest launched his object from atop the playset.

Ann climbed a tall tree to launch her object.

Damian’s object went the highest.

Debbie’s object was in the air the shortest amount of time.

The arrow was shot by the oldest.

The golf ball went 87 feet into the air.

Ann shot her arrow higher than the tennis and bowling balls.

The bowling ball reached its maximum height just under 1 second.

Damian’s ball was in the air twice as long as the 2nd oldest sibling.

The object of the youngest brother was in the air for more than 4 seconds. Relevance

"Ann climbed a tall tree to launch her object."

Ann's equation must be y₃ ◄

owing to the "60" term at x = 0.

"The arrow was shot by the oldest."

"Ann shot her arrow..."

That makes Ann the oldest and gives her the arrow. ◄

The rest require some work. The first derivatives are

dy₁/dx = -32x + 62

and so for dy₁/dx = 0 (maximum height), x = 62/32

at which point y₁ = -16(62/32)² + 62*62/32 + 10 = 70 ft

dy₂/dx = -32 + 75

and so for dy₂/dx = 0, x = 75/32

at which point y₂ = -16(75/32)² + 75*75/32 = 88 ft

dy₃/dx = -32x + 38

and so for dy₃/dx = 0, x = 38/32

at which point y₃ = -16*(38/32)² + 38*38/32 + 60 = 83 ft

I can't make a subscript 4. For y4,

dy/dx = -32x + 20

and so for dy/dx = 0, x = 20/32

at which point y = -16(20/32)² + 20*20/32 + 10 = 16 ft

y4 is the bowling ball, having the lowest height.

"Damian’s object went the highest."

So Damian's equation is y₂ ◄

and he must have lobbed the golf ball.

"Debbie’s object was in the air the shortest amount of time."

and so Debbie's equation must be y4 ◄

the bowling ball having gained only 16 feet of altitude.

That leaves y₁ for Adam. ◄

"Ann shot her arrow higher than the tennis and bowling balls."

Knowing the bowling ball is Debbie's, that leaves the tennis ball and equation y₁ which we know to be Adam's. ◄

Only now does it occur to me that "x" is actually the time here rather than the horizontal distance. Then solving these quadratics for y = 0 gives the time of flight. The results:

t₁ = 4.0 s

t₂ = 4.7 s

t₃ = 3.5 s

t4 = 1.6 s

"The object of the youngest brother was in the air for more than 4 seconds."

t₁ and t₂ both qualify.

"The youngest launched his object from atop the playset."

That must correspond to either y₁ or y4.

But combining these last two means that t₁ corresponds to the youngest brother, and that means Adam. ◄

"Damian’s ball was in the air twice as long as the 2nd oldest sibling."

Since only Damian and Debbie are left, that makes Debbie "the 2nd oldest sibling." ◄

But his ball flight time was around three times Debbie's.

Results:

Ann ............ y₃ ....... arrow ............. oldest

Debbie ....... y4 ...... bowling ball ... 2nd oldest

Damian ...... y₂ ....... golf ball .........

Adam ......... y₁ ....... tennis ball ...... youngest