need help with math question

Question

A company finds they price thier pens at p dollars each, then thier weekly revenue for the pens will be 1000p - 200p^2 dollars. If their weekly revenue is 1200 dollars what could be the price in dollars of each pen

Answer 1

Hi Imelda!

The price of pens would either be $2 or $3. Let me show you how I approached this problem!

1. Since we know the equation 1000*p-200*p² finds the weekly revenue and that $1200 is the weekly revenue, we can set the equation equal to $1200.
2. This results in 1000*p-200*p²=1200
3. Lets simplify this equation by moving everything to one side and by dividing each term by their greatest common factor (the greatest common factor of 1000, -200, and 1200 is 200).
4. First moving all of the terms to one side: 0=200*p²-1000*p+1200
5. And now simplify by dividing each term by their greatest common factor (200): 0=p²-5*p+6
6. Now lets factor the quadratic equation
7. This yields 0=(p-3)*(p-2)
8. Since the quadratic equation must equal 0, either (p-3) or (p-2) must equal 0.
9. Thus, we can make the equations: (p-3) = 0 or (p-2)=0
10. Solving this yields that p=3 or p=2

Therefore, the price of the pens could either be $2 or $3.

Feel free to ask me more questions if this doesn't make sense!

I hope this helps :)

Manu

Answer 2

The fact we are given can be written as an equation. We can detect that the word "be" is equivalent to having an equals sign.

Let us call one week's revenue (in dollars) R.

The equation is then : R = 1000 * p - 200 p²

The question asks us to find p, the price of the pens ( in dollars per pen), if R = 1200$

Subbing in the value proposed in the question for R, we have:

1200 = 1000 * p - 200 p²

We recognize that the right side is a polynomial in the unknown p and that the overall equation is a quadratic equation because there is a term in which the exponent of the unknown variable is equal to 2 and this 2 is the highest degree term and all of the terms have the unknown variable to some non-negative integer power.

To solve a quadratic we move all terms to the same side (leaving zero on one side). We do this by adding 200 p² to both sides and subtracting 1000*p from both sides. It is also standard to write the terms in order of decreasing power.

200 p² - 1000 p + 1200 = 0

We can divide both sides by 200 to simplify things.

p² - 5 p + 6 = 0

Multiplying the first and last coefficient, we get 6 so if we can find a factor pair of 6 that multiplies to give 36 but adds to give -5 (the middle coefficient) we can use that to break up -5p into two parts.

We need two factors of the same sign to produce the positive product, so they must be both be negative to produce a negative sum.

-1&-6? No, because the sum would be -7.

-2 & -3? Yes, because the sum is -5.

Therefore p² - 5 p + 6 = (p - 3) (p -2 ) = 0

(

OR :

Factoring out a p from the first two terms and a negative 2 from the latter two,

p (p-3) + (-2)( p - 3) = 0

Factoring out the common factor of p-3,

(p - 3) (p -2 ) = 0

)

There are two possible solutions to this equation because there are two factors whose product is zero.

There are two possible ways for the equation to be true. Either the first factor is equal to zero or the second factor is. We must consider these possibilities separately to find that:

p =3 or p =2.

So the price in dollars of each pen (according to the given model) is either $2 per pen or $3 per pen.

(Note that there is a sweet spot midway between these values at which revenue would have been higher, according to the model we were told to use!)