This problem involves calculating the combined work rate of multiple taps (inlets and outlets) to determine the total time taken to fill a tank. We will calculate the individual work rates of each tap (fraction of the tank filled or emptied per hour) and then sum them up to find the net work rate. Finally, the reciprocal of the net work rate will give the total time.
Calculate the filling rate of tap A:
Tap A fills the tank in 10 hours. So, in 1 hour, tap A fills \(\frac{1}{10}\) of the tank.
Calculate the filling rate of tap B:
Tap B fills the tank in 14 hours. So, in 1 hour, tap B fills \(\frac{1}{14}\) of the tank.
Calculate the emptying rate of the third tap:
The third tap empties the tank in 35 hours. So, in 1 hour, the third tap empties \(\frac{1}{35}\) of the tank. This rate will be subtracted from the filling rates.
Calculate the net filling rate when all taps are open:
When all three taps are turned on simultaneously, the net fraction of the tank filled in 1 hour will be the sum of the filling rates minus the emptying rate:
\[ \text{Net rate} = \text{Rate of A} + \text{Rate of B} - \text{Rate of emptying tap} \] \[ \text{Net rate} = \frac{1}{10} + \frac{1}{14} - \frac{1}{35} \]To add and subtract these fractions, find a common denominator. The least common multiple (LCM) of 10, 14, and 35 is 70.
\[ \text{Net rate} = \frac{7}{70} + \frac{5}{70} - \frac{2}{70} \] \[ \text{Net rate} = \frac{7 + 5 - 2}{70} \] \[ \text{Net rate} = \frac{12 - 2}{70} \] \[ \text{Net rate} = \frac{10}{70} \] \[ \text{Net rate} = \frac{1}{7} \]So, in 1 hour, \(\frac{1}{7}\) of the tank is filled.
Calculate the total time to fill the tank:
If \(\frac{1}{7}\) of the tank is filled in 1 hour, then the entire tank will be filled in the reciprocal of this rate.
\[ \text{Time} = \frac{1}{\text{Net rate}} \] \[ \text{Time} = \frac{1}{\frac{1}{7}} \] \[ \text{Time} = 7 \text{ hours} \]D) 7
The calculations show that the tank will be completely filled in 7 hours when all three taps are operating simultaneously.
This question asks us to evaluate the strength of two arguments regarding whether the government should stop allowing new industries. We need to determine which argument presents a more compelling and holistic viewpoint for national policy decisions.
Correct Option: B) Only argument II is strong
Argument II presents a more comprehensive and vital perspective for national development. While environmental concerns (Argument I) are valid, they are usually addressed through regulations and technological advancements rather than a complete halt to industrialization, which would severely impede economic progress and employment opportunities.
What is the square root of (x2 + 4x + 4)(x2 + 6x + 9)?
(2x + 3) (x + 3)
(x + 2) (x + 3)
(x + 2) (2x + 3)
(x + 2) (x + 4)
To find the square root of the given expression, we need to factorize each quadratic term into a perfect square. We will use the algebraic identity \((a+b)^2 = a^2 + 2ab + b^2\).
The first term is \(x^2 + 4x + 4\).
We can recognize this as a perfect square trinomial. Comparing it with \((a+b)^2 = a^2 + 2ab + b^2\), we have \(a=x\) and \(b=2\).
So, \(x^2 + 4x + 4 = x^2 + 2(x)(2) + 2^2 = (x+2)^2\).
The second term is \(x^2 + 6x + 9\).
Similarly, this is also a perfect square trinomial. Comparing it with \((a+b)^2 = a^2 + 2ab + b^2\), we have \(a=x\) and \(b=3\).
So, \(x^2 + 6x + 9 = x^2 + 2(x)(3) + 3^2 = (x+3)^2\).
The original expression is \((x^2 + 4x + 4)(x^2 + 6x + 9)\).
Substituting the factored forms, we get \((x+2)^2 (x+3)^2\).
We need to find the square root of \((x+2)^2 (x+3)^2\).
\[ \sqrt{(x+2)^2 (x+3)^2} = \sqrt{(x+2)^2} \cdot \sqrt{(x+3)^2} \]\[ = |x+2| \cdot |x+3| \]Assuming \(x\) is such that \(x+2\) and \(x+3\) are non-negative (which is typically implied in such problems unless specified otherwise, or we consider the principal square root), we can write:
\[ = (x+2)(x+3) \]B) \((x + 2) (x + 3)\) is the correct square root of the given expression, as derived in the step-by-step solution.
The question asks to identify the Indian film that won the Global Sustainability Film Award in London in 2017. This requires knowledge of specific film awards and their recipients.
Correct Option: A) Lipstick Under My Burkha
"Lipstick Under My Burkha" won the Global Sustainability Film Award at the London Asian Film Festival in 2017. The film, directed by Alankrita Shrivastava, explores the secret lives and desires of four women in a small Indian town, challenging societal norms and patriarchal structures. Its themes of female empowerment and breaking free from oppression align with the spirit of sustainability in a social context.
This problem involves a direct proportionality relationship between the number of workers and the number of candles produced when the time is constant. We can use the concept of work done, where the total work is the number of candles. The efficiency of each worker is assumed to be constant.
Identify the given information:
Understand the relationship: If the time is constant, the number of workers is directly proportional to the number of candles produced. This means if more candles need to be made, more workers are required.
Set up the proportion:
The ratio of workers to candles should remain constant if the time is the same. So,
\[ \frac{W_1}{C_1} = \frac{W_2}{C_2} \]Substitute the known values into the proportion:
\[ \frac{12}{111} = \frac{W_2}{148} \]Solve for \(W_2\):
To find \(W_2\), multiply both sides by 148:
\[ W_2 = \frac{12}{111} \times 148 \]Perform the calculation:
First, simplify the fraction \(\frac{12}{111}\). Both numbers are divisible by 3:
\[ \frac{12 \div 3}{111 \div 3} = \frac{4}{37} \]Now, substitute this back into the equation for \(W_2\):
\[ W_2 = \frac{4}{37} \times 148 \]Notice that 148 is a multiple of 37. \(37 \times 4 = 148\).
So,
\[ W_2 = 4 \times 4 \]\[ W_2 = 16 \]Therefore, 16 people will have to be engaged to make 148 candles in the same time.
Correct Option: C) 16 people
Acceleration due to gravity g = ________
GMR
\(\frac{MR^2}{G}\)
\(\frac{G}{MR^2}\)
\(\frac{GM}{R^2}\)
The acceleration due to gravity, denoted by \(g\), is the acceleration experienced by an object due to the gravitational pull of a celestial body (like Earth). It can be derived from Newton's Law of Universal Gravitation and Newton's Second Law of Motion.
Newton's Law of Universal Gravitation: This law states that the gravitational force (\(F\)) between two objects is directly proportional to the product of their masses and inversely proportional to the square of the distance between their centers. For an object of mass \(m\) on the surface of a planet of mass \(M\) and radius \(R\), the gravitational force is given by:
\[ F = \frac{GMm}{R^2} \]where \(G\) is the universal gravitational constant.
Newton's Second Law of Motion: This law states that the force acting on an object is equal to its mass times its acceleration (\(F = ma\)). In the case of gravity, the acceleration is the acceleration due to gravity, \(g\). So, the gravitational force can also be expressed as:
\[ F = mg \]Equating the Forces: By equating the two expressions for the gravitational force, we can find the formula for \(g\):
\[ mg = \frac{GMm}{R^2} \]Solving for \(g\): We can cancel out the mass of the object (\(m\)) from both sides of the equation:
\[ g = \frac{GM}{R^2} \]This formula shows that the acceleration due to gravity depends on the mass of the planet (\(M\)), its radius (\(R\)), and the universal gravitational constant (\(G\)), but not on the mass of the object experiencing the gravity.
D) \(\frac{GM}{R^2}\) is the correct formula for the acceleration due to gravity, derived from Newton's Law of Universal Gravitation and Newton's Second Law of Motion.
The question asks about the relative thickness of the ventricular wall compared to the atrial wall in the heart. Understanding the function of each chamber provides the key to answering this.
Correct Option: A) The wall of the ventricle is thick from the wall of the atrium. This is because ventricles need to pump blood over longer distances and against higher pressure compared to atria.
The speed of sound in a medium depends on its elasticity (stiffness) and density. Generally, sound travels faster in solids than in liquids, and faster in liquids than in gases. Within solids, the speed is higher in materials that are more elastic and less dense. We need to compare the properties of the given metals to determine where the speed of sound will be maximum.
where \(Y\) is the Young's Modulus (a measure of elasticity or stiffness) and \(\rho\) is the density of the material.
Note: These values can vary slightly depending on the specific alloy and temperature, but the general order holds.
Comparing these approximate speeds, Aluminium has the highest speed of sound among the given options. While Steel has a higher Young's Modulus, Aluminium's significantly lower density gives it a higher speed of sound.
C) Aluminium — Based on the calculation, Aluminium has the highest speed of sound among the given options due to its favorable ratio of Young's Modulus to density.
The area of a triangle can be calculated using the formula \[\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}\]. Since the area of a given triangle is constant, the height from a vertex is inversely proportional to the length of the side it is drawn to. We will use this relationship to find the ratio of the heights.
Let the sides of the triangle be \(a = 6 \text{ cm}\), \(b = 11 \text{ cm}\), and \(c = 13 \text{ cm}\).
Let the respective heights from the opposite vertices be \(h_a\), \(h_b\), and \(h_c\).
The area of the triangle, denoted by \(A\), can be expressed in three ways:
\[ A = \frac{1}{2} a h_a \]\[ A = \frac{1}{2} b h_b \]\[ A = \frac{1}{2} c h_c \]From these equations, we can write:
\[ 2A = a h_a = b h_b = c h_c \]This implies that the product of a side and its corresponding height is constant. Therefore, the heights are inversely proportional to the lengths of the sides:
\[ h_a : h_b : h_c = \frac{1}{a} : \frac{1}{b} : \frac{1}{c} \]Substitute the given side lengths:
\[ h_a : h_b : h_c = \frac{1}{6} : \frac{1}{11} : \frac{1}{13} \]To remove the fractions, multiply each term by the Least Common Multiple (LCM) of the denominators (6, 11, 13). Since 6, 11, and 13 are coprime (11 and 13 are prime, 6 is \(2 \times 3\)), their LCM is their product:
\[ \text{LCM}(6, 11, 13) = 6 \times 11 \times 13 = 66 \times 13 = 858 \]Multiply each term in the ratio by 858:
\[ h_a : h_b : h_c = \frac{1}{6} \times 858 : \frac{1}{11} \times 858 : \frac{1}{13} \times 858 \]\[ h_a : h_b : h_c = 143 : 78 : 66 \]C) 143 : 78 : 66 is the correct ratio of the heights.