Quadratic word problems (standard form)

1. Fish and water temperature

The number of fish is a function of the water's temperature:

pip install sympy

16.9s

Bash in Python

Which temperatures will result in fish?

import sympy as sp 
import matplotlib.pyplot as plt
import numpy as np

def plot(a, b, c, xax, yax):
  x = sp.Symbol('x')
  roots = sorted(sp.solveset(a*x**2 + b*x + c))
  xaxis = np.arange(roots[0], roots[1], 0.01)
  yaxis = a*xaxis**2 + b*xaxis + c
  fig, ax = plt.subplots()
  ax.plot(xaxis, yaxis)
  ax.set(xlabel=xax, ylabel=yax)
  ax.grid()
  return;

plot(-2, 40, -72, 'Temperature', 'Fish')
plt.gcf()

2.3s

Python

There are no fish when .

To find the zeros, we need to factor the equation as a product of 2 binomials. First divide by to get by itself:

(defn divide-by-a [[a b c r]]
  [1 (/ b a) (/ c a) (/ r a)])

(defn div-by-a [s]
  (map #(/ % (first s)) s))

(div-by-a [-2 40 -72 0])

0.2s

Clojure

List(4) (1, -20, 36, 0)

We want to find the positive and negative integer factors of :

(defn factors [n]
  (for [x (into (range (- (Math/abs n)) 0)
                (range 1 (inc (Math/abs n))))
        :when (zero? (rem n x))]
    [(int x) (int (/ n x))]))

(factors 0)

0.2s

Clojure

List(0) ()

Now find the ones that add up to and solve for by taking their negatives:

(defn zeros [[a b c r]]
  (map - (first (filter #(= b (apply + %))
                        (factors c)))))


(-> [-1 100 0 0]
  divide-by-a)

(zeros [1 -100 0 0])

0.1s

Clojure

List(0) ()

2. Profit and price of an item

Profit is a function of the price of an item:

plot(-2, 16, -24, 'Price', 'Profit')
plt.gcf()

1.8s

Python

What price results in the maximum profit?

Find the vertex's -coordinate by taking the average of the two zeros:

(defn average [x y]
  (/ (+ x y) 2))

(defn vert-x [[a b c r]]
  (apply average
       (-> [a b c r]
         divide-by-a
         zeros)))

(vert-x [-2 16 -24 0])

0.1s

Clojure

3. Height of a thrown stone

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

The stone's height, seconds after Alain threw it, is modeled by:

plot(-5, 10, 15, 'Seconds', 'Height')
plt.gcf()

1.8s

Python

What is the height of the stone at the time it is thrown?

That's easy, substitute for to get .

4. Height after takeoff

The spaceship's height (in meters), seconds after takeoff, is modeled by:

plot(-2, 20, 48, 'Seconds', 'Height')
plt.gcf()

1.7s

Python

What is the maximum height that the ship will reach?

(vert-x [-2 20 48 0])

0.1s

Clojure

The vertex's -coordinate is .

Now find :

(defn solve [[a b c r] x]
  (+ (* (* x x) a)
     (* b x)
     c))

(solve [-2 20 48 0] 5)

0.1s

Clojure

5. Object launched from platform

Its height, seconds after the launch, is modeled by:

How many seconds after launch will the object land on the ground?

(->> [-5 20 60 0]
  divide-by-a
  zeros
  (apply max))

0.0s

Clojure

What is the height of the object at the time of launch?

(solve [-5 20 60 0] 0)

0.0s

Clojure

6. Power generated from circuit

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

What is the maximum power generated by the circuit?

To find the roots, we need to get it into factored form.

Divide by a:

(div-by-a [-12 120 0])

0.1s

Clojure

List(3) (1, -10, 0)

Divide by the variable and take the root of each term:

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

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

(defn factor-x [[a b c]]
  [[1 0] [1 (/ b a)]])

(apply roots (factor-x [1 -10 0]))

0.1s

Clojure

Vector(2) [0, 10]

The vertex's coordinate is the average of the 2 roots; which we will then plug into the equation for the coordinate:

(defn vertex [[a b c :as eq]]
  (let [div-a (div-by-a eq)
        roots (apply roots (factor-x div-a))
        x (apply average roots)
        y (+ (* (* x x)
                a)
             (* b x)
             c)]
    [x y]))

(vertex [-12 120 0])

0.1s

Clojure

Vector(2) [5, 300]

7. The garden's area

The garden's area is a function of the garden's width :

What side width will produce the maximum garden area?

(vertex [-1 100 0])

0.0s

Clojure

Vector(2) [50, 2500]

8. Ball thrown off balcony

The ball's height (in meters above the ground), seconds after being thrown, is modeled by:

How many seconds after being thrown will the ball hit the ground?

(->> [-2 4 16]
  div-by-a
  zeros
  (apply max))

0.1s

Clojure

9. Diving altitude

A diver's altitude (in meters relative to sea level), seconds after diving, is modeled by:

How many seconds after diving will the diver reach the lowest altitude?

(vertex [1/2 -10 0])

0.2s

Clojure

Vector(2) [#n"10", #n"-50"]