Two water taps together can fill a tank in $3\frac{3}{13}$ hours. The tap of larger diameter takes 5 hours less than the smaller one to fill the tank separately. Find the time in which each tap can fill the tank separately.

Step 1: Define variables and rates

Let $t_S$ be the time taken by the smaller tap to fill the tank alone. Since the larger tap takes 5 hours less, its time will be $t_S - 5$. The rate at which a tap fills the tank is the reciprocal of the time it takes. So, the rate of the smaller tap is $1/t_S$ and the rate of the larger tap is $1/(t_S - 5)$.

Step 2: Formulate the combined rate equation

When both taps work together, their rates add up. The combined time to fill the tank is given as $3\frac{3}{13}$ hours, which is $42/13$ hours. Therefore, the combined rate is $13/42$. We set up the equation by summing the individual rates and equating it to the combined rate.

Step 3: Simplify the equation to a quadratic form

First, combine the fractions on the left side of the equation by finding a common denominator. Then, cross-multiply to eliminate the denominators. Expand and rearrange the terms to form a standard quadratic equation in the form $ax^2 + bx + c = 0$.

Step 4: Solve the quadratic equation for $t_S$

We use the quadratic formula $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ to solve for $t_S$. Substitute the coefficients $a=13$, $b=-149$, and $c=210$ into the formula. Calculate the discriminant and then find the two possible values for $t_S$.

Step 5: Determine the valid time for each tap

We have two possible values for $t_S$. Since time cannot be negative, we must discard the solution that leads to a negative time for the larger tap. If $t_S = 1.65$ hours, then $t_L = 1.65 - 5 = -3.35$ hours, which is not physically possible. Thus, $t_S = 9.81$ hours is the correct time for the smaller tap, and $t_L = 9.81 - 5 = 4.81$ hours for the larger tap.

More from this chapter

• find two natural numbers the sum of 4 squares is 25 times the sum and also equal to 3 times the difference
• Split middle term and factorize X square + 32 x minus 420 is equal 0
• divide 16 into 2 parts such that the twice the square of the largest part exceeds the square of the smaller part by 164
• एक्स स्क्वायर + 24 + 12 सॉल्व करो
• Solve: x² − 5x + 6 = 0 vdhsjv
• Solve: x² − 5x + 6 = l don't know
• 19. The perimeter of a right triangle is 60 cm and its hypotenuse is 25 cm. Find the lengths of other two sides of the triangle.
• The time taken by a person to travel an upward distance of 150 km was 2(1)/(2) hours more than the time taken in the downward return journey. If he returned at

All Quadratic Equations solutions →