Quadratic word problems (factored form)

Amir stands on a balcony and throws a ball to his dog, who is at ground level.

The ball's height (in meters above the ground), seconds after Amir threw it, is modeled by

from sympy import *

x = Symbol('x')

coeff = -1
term1 = 1
term2 = -7
y = coeff * (x + term1)*(x + term2)
zero1 = solveset(Eq(x + term1))
zero2 = solveset(Eq(x + term2))
vertx = sum(zero1 + zero2) / 2
verty = y.subs(x, vertx)
vertex = (vertx, verty);
graph = {'Vertex': vertex, 'Zeros': zero1 + zero2}
graph

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nil

What is the maximum height that the ball will reach?

verty

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nil

How many seconds after being thrown will the ball reach its maximum height?

vertx

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nil

The ball's height is modeled by a quadratic function, whose graph is a parabola.

The maximum height is reached at the vertex.

So in order to find the maximum height, we need to find the vertex's -coordinate.

We will start by finding the vertex's -coordinate, and then plug that into .

The vertex's -coordinate is the average of the two zeros, so let's find those first:

Now let's take the zeros' average:

The vertex's -coordinate is . Now let's find :

In conclusion, the maximum height that the ball will reach is meters.

Ana dives into a pool off of a springboard high dive.

Her height (in meters above the water), seconds after diving, is modeled by

h(x)=-5(x+1)(x-3)

What is the height of Ana above the water at the time of diving?

-5*(0 + 1)*(0 - 3)

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nil

How many seconds after diving will Ana hit the water?

Ana hits the water when .

We found that for or . Since doesn't make sense in our context, the only reasonable answer is .

In conclusion, Ana hit the water after seconds.

Simon has meters of fencing to build a rectangular garden.

The garden's area (in square meters) as a function of the garden's width (in meters) is modeled by

What width will produce the maximum garden area?

The maximum area is reached at the vertex. So in order to find the width that produces the maximum area, we need to find the vertex's -coordinate.

(defn root [[b c]]
  (- (/ c b)))

(defn roots [eq1 eq2]
  [(root eq1) (root eq2)])

(defn vertex [n f1 f2]
  (let [x (/ (+ (root f1) (root f2)) 2)
        y (* n
             (+ (* (first f1) x)
                (last f1))
             (+ (* (first f2) x)
                (last f2)))]
    [x y]))

{:vertex (vertex 1 [-1 0] [1 -80])
 :roots (roots [-1 0] [1 -80])}

0.2s

Clojure

Map {:vertex: Vector(2), :roots: Vector(2)}

In conclusion, the maximum area is produced when the rectangle's width is meters.

What is the maximum area possible?

In order to find the maximum area, we need to find the vertex's -coordinate.

Guillermo is a professional deep water free diver.

His altitude (in meters relative to sea level), seconds after diving, is modeled by

What is the lowest altitude Guillermo will reach?

To find the lowest altitude, we need to find the vertex's -coordinate.

(vertex 1 [1/20 0] [1 -100])

0.1s

Clojure

Vector(2) [#n"50", #n"-125"]

In conclusion, the lowest altitude Guillermo will reach is meters relative to sea level.

A certain company's main source of income is a mobile app.

The company's annual profit (in millions of dollars) as a function of the app's price (in dollars) is modeled by

What would be the company's profit if the app price is dollars?

The company's profit if the app price is dollars is given by .

In conclusion, the company will have a profit of million dollars if the app price is dollars.

(In other words, the company would lose million dollars.)

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Python

Ricardo throws a stone off a bridge into a river below.

The stone's height (in meters above the water), seconds after Ricardo threw it, is modeled by

How many seconds after being thrown will the stone reach its maximum height?

(vertex -5 [1 -8] [1 4])

0.1s

Clojure

Vector(2) [2, 180]

The stone will reach its maximum height after seconds.

(-1 + 7) / 2

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nil

An object is launched from a platform.

Its height (in meters), seconds after the launch, is modeled by

How many seconds after launch will the object hit the ground?

The object hits the ground when .

(roots [1 1] [1 -9])

0.1s

Clojure

Vector(2) [-1, 9]

We found that for or . Since doesn't make sense in our context, the only reasonable answer is .

Therefore, the object will hit the ground after seconds.

A hovercraft takes off from a platform.

Its height (in meters), seconds after takeoff, is modeled by

What is the height of the hovercraft at the time of takeoff?

The height of the hovercraft at the time of takeoff is given by .

(defn f-of [n f1 f2 x]
  (+ (* n
        (+ (* (first f1) x)
           (last f1))
        (+ (* (first f2) x)
           (last f2)))))

(f-of -1 [1 -11] [1 3] 0)

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In conclusion, the height of the hovercraft at the time of takeoff is meters.

How many seconds after takeoff will the hovercraft land on the ground?

The hovercraft lands on the ground when .

(roots [1 -11] [1 3])

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Clojure

Vector(2) [11, -3]

We found that for or . Since doesn't make sense in our context, the only reasonable answer is .

In conclusion, the hovercraft will land on the ground after seconds.

The power generated by an electrical circuit (in watts) as a function of its current (in amperes) is modeled by

What current will produce the maximum power?

To find the current that will produce the maximum power, we need to find the vertex's -coordinate.

(vertex 1 [-15 0] [1 -8])

0.1s

Clojure

Vector(2) [4, 240]

In conclusion, the maximum power occurs when the current is amperes.