15. A tank fills completely in 2 hours if both the taps are open. If only one of the taps is open at the given time, the smaller tap takes 3 hours more than the larger one to fill the tank. How much time does each tap take to fill the tank completely ?
Answers
Answer:
6.
Step-by-step explanation:
Let x be the hours taken by the larger tap and then x+3 is the time taken by the smaller tap
to fill the tank
The amount of water filled by the larger tap in one hour =
x
1
The amount of water filled by the smaller tap in one hour =
x+3
1
We know that, both the tap fill the tank in 2 hours if they are open together.
According to the condition,
x
1
+
x+3
1
=
2
1
⇒
x
2
+3x
2x+3
=
2
1
⇒4x+6=x²+3x
⇒x²-x-6=0
Step 2: Solving the quadratic equation
⇒x²
−3x+2x−6=0
[Using middle term split method]
⇒x(x−3)+2(x−3)=0
⇒(x−3)(x+2)=0
⇒(x−3)=0or(x+2)=0
x=3,−2
∴x=3 [Taking only positive value as time can’t be negative]
Thus, the larger tap takes 3 hours to fill the tank and the smaller tap takes [3+3]=6
hours to fill the tank.
Step-by-step explanation:
Let the larger tap take x hours to fill the tank completely. ∴ Part of tank filled by the larger tap in 1 hour = 1/x Also, the smaller tap takes (x + 3) hours to fill the tank completely. ∴ Part of tank filled by the smaller tap in 1 hour = 1/(x+3) ∴Part of tank filled by both the taps in 1 hour = ( 1/x + 1/(x + 3) But, the tank gets filled in 2 hours by both the taps. ∴ Part of tank filled by both the taps in 1 hour =1/2 According to the given condition, 1/x + 1/(x + 3) = 1/2 ∴ (x + 3 + x)/x(x + 3) = 1/2 ∴ (2x + 3)/x(x + 3) = 1/2 ∴ 2(2x + 3) = x(x + 3) ∴ 4x + 6 = x2 + 3x ∴ x2 + 3x – 4x – 6 = 0 ∴ x2 – x – 6 = 0 ∴ x2 – 3x + 2x – 6 = 0 ∴ x(x – 3) + 2(x – 3) = 0 ∴ (x – 3)(x + 2) = 0 By using the property, if the product of two numbers is zero, then at least one of them is zero, we get ∴ x – 3 = 0 or x + 2 = 0 ∴ x = 3 or x = -2 But, time cannot be negative. ∴ x = 3 and x + 3 = 3 + 3 = 6 ∴ The larger tap takes 3 hours and the smaller tap takes 6 hours to fill the tank completely.Read more on Sarthaks.com - https://www.sarthaks.com/844685/tank-fills-completely-hours-both-taps-open-only-the-taps-open-the-given-time-the-smaller-tap