1. ## More QE

I need help solving these three problems using a quadratic equation:

Two taps turned on together can fill a tank in 15 minutes. One tap alone takes 16 minutes longer than the other to fill the tank. Find the time it takes for each tap to fill the tank on its own.

Pauline would take 4 hours longer than Dora to paint their 2-bedroom apartment. Together they can get the job done in 4.8 hours. How long would Dora take to paint the apartment by herself?

I am thinking of three numbers that are squares of consecutive integers. Twice the smallest number minus three times the middle one plus four times the largest one comes to a total of 230. What are my three numbers?

Any help is appreciated greatly! thankks : )

2. Hello, foreverbrokenpromises!

Here's the last one . . .

I am thinking of three numbers that are squares of consecutive integers.
Twice the smallest number minus three times the middle one plus four times the largest totals 230.
What are my three numbers?

Let the squares be: .$\displaystyle x^2,\;(x+1)^2,\;(x+2)^2$

. . $\displaystyle \underbrace{\text{2 times smallest}}_{2x^2} \: \underbrace{\text{minus}}_{-} \: \underbrace{\text{3 times middle}}_{3(x+1)^2} \: \underbrace{\text{plus}}_{+} \: \underbrace{\text{4 times largest}}_{4(x+2)^2} \: \underbrace{\text{totals}}_{=} \: \underbrace{230}_{230}$

We have: .$\displaystyle 2x^2 - 3(x+1)^2 + 4(x+2)^2 \;=\;230$

Expand: .$\displaystyle 2x^2 - 3x^2 - 6x - 3 + 4x^2 + 16x + 16 \:=\:230$

Simplify: .$\displaystyle 3x^2 + 10x - 217 \:=\:0$

Factor: .$\displaystyle (x-7)(3x+31) \:=\:0$

. . which has the integer solution: .$\displaystyle x \:=\:7$

Hence, the three consecutive integers are: .$\displaystyle 7,8,9$

Therefore, the three squares are: .$\displaystyle \boxed{49,\:64,\:81}$

3. Originally Posted by foreverbrokenpromises
I need help solving these three problems using a quadratic equation:

Two taps turned on together can fill a tank in 15 minutes. One tap alone takes 16 minutes longer than the other to fill the tank. Find the time it takes for each tap to fill the tank on its own.
When you have two things or people working together, their rates add. Let "T" be the time required for the "faster" tap to fill the tank, in minutes. Then the time required for the slower tap is T+ 16 minutes. The faster tap has a rate of 1 tank/T min= 1/T and the slower tap has a rate of 1 tank/(T+ 16 minutes)= 1/(T+16). Working together they have a rate of $\displaystyle \frac{1}{T}+ \frac{1}{T+ 16}= \frac{T+15}{T(T+16)}+ \frac{T}{T(T+ 16)}$$\displaystyle = \frac{2T+16}{T(T+16}$ "tanks per minutes". The time to fill one tank 1 tank*$\displaystyle \frac{T(T+16)}{2T+ 16}$ "minutes per tank"= $\displaystyle \frac{T(T+16)}{2T+ 16}= 15$
Multiply both sides by the denominator, T(T+ 16) and you have T(T+16)= 15(2T+ 16) or $\displaystyle T^2+ 16T= 30T+ 240$ which reduces to $\displaystyle T^2- 14T- 240= 0$.

Can you solve that for T and then add 16 for the second value?

Pauline would take 4 hours longer than Dora to paint their 2-bedroom apartment. Together they can get the job done in 4.8 hours. How long would Dora take to paint the apartment by herself?
Pretty much the same as the previous one. Let T be the time it takes Dora to paint the appartment. Then the time it take Pauline is T+ 4. Their respective rates are $\displaystyle \frac{1}{T}$ "apartments per hour" and $\displaystyle \frac{1}{T+4}$ "apartments per hour". Working together they paint at the rate of $\displaystyle \frac{1}{T}+ \frac{1}{T+4}$ "apartments per hour" and the reciprocal is "hours per apartment" when they are working together. Set that equal to 4.8 and solve for T.

I am thinking of three numbers that are squares of consecutive integers. Twice the smallest number minus three times the middle one plus four times the largest one comes to a total of 230. What are my three numbers?