The angle of depression of the foot of a building from the top of a tower 36 m away is 30°. What is the height of the tower?
This problem involves trigonometry, specifically the concept of the angle of depression. We can visualize the situation as a right-angled triangle where the height of the tower is one side, the distance to the building is the other side, and the line of sight forms the hypotenuse. We will use the tangent function to relate the angle of depression, the height of the tower, and the distance from the tower to the building.
Let AB be the tower and C be the foot of the building. The distance between the tower and the building is BC = 36 m. Let the height of the tower be AB = \(h\) meters.
The angle of depression from the top of the tower (A) to the foot of the building (C) is 30°. This means if we draw a horizontal line AX from A, then \(\angle XAC = 30^\circ\).
Since AX is parallel to BC, the alternate interior angle \(\angle ACB = \angle XAC = 30^\circ\).
Consider the right-angled triangle \(\triangle ABC\), where the right angle is at B.
In \(\triangle ABC\), we know the base BC and the angle \(\angle ACB\). We need to find the perpendicular side AB (height of the tower).
The tangent of an angle in a right-angled triangle is given by:
\[ \tan(\theta) = \frac{\text{Opposite side}}{\text{Adjacent side}} \]Here, \(\theta = 30^\circ\), the opposite side is AB (\(h\)), and the adjacent side is BC (36 m).
\[ \tan(30^\circ) = \frac{AB}{BC} \]We know that \(\tan(30^\circ) = \frac{1}{\sqrt{3}}\).
\[ \frac{1}{\sqrt{3}} = \frac{h}{36} \]Now, solve for \(h\):
\[ h = \frac{36}{\sqrt{3}} \]To rationalize the denominator, multiply the numerator and denominator by \(\sqrt{3}\):
\[ h = \frac{36}{\sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}} = \frac{36\sqrt{3}}{3} \] \[ h = 12\sqrt{3} \text{ m} \]Correct Option: A) 12\(\sqrt{3}\) m
The question asks to identify which subatomic particle is responsible for constituting an electric current. An electric current is defined as the flow of electric charge. Therefore, we need to determine which of the given particles are charged and are typically free to move in a conductor.
D) Electrons — Electrons are negatively charged particles that are relatively light and, in conductive materials, are free to move. Their directed movement constitutes an electric current.
A box contains 286 pencils to be distributed among three children in the ratio of \( \frac{1}{2} \) : \( \frac{1}{3} \) : \( \frac{1}{4} \). Find the number of pencils received by the first child.
To distribute items in a given ratio of fractions, first convert the fractional ratio into a simple whole number ratio by finding the Least Common Multiple (LCM) of the denominators. Then, calculate the total parts of the ratio and use this to find the share of each child.
Convert the fractional ratio to a simple ratio:
The given ratio is \[\frac{1}{2} : \frac{1}{3} : \frac{1}{4}\].
Find the LCM of the denominators (2, 3, 4). The LCM of 2, 3, and 4 is 12.
Multiply each fraction in the ratio by the LCM:
\[ \frac{1}{2} \times 12 : \frac{1}{3} \times 12 : \frac{1}{4} \times 12 \] \[ 6 : 4 : 3 \]So, the simple ratio of distribution is \( 6:4:3 \).
Calculate the total parts of the ratio:
Add the parts of the simple ratio: \( 6 + 4 + 3 = 13 \).
The total number of pencils is 286.
Find the number of pencils received by the first child:
The first child's share corresponds to the first part of the ratio, which is 6.
Number of pencils for the first child \[= \frac{\text{First child's ratio part}}{\text{Total ratio parts}} \times \text{Total pencils}\]
\[ \text{Number of pencils for first child} = \frac{6}{13} \times 286 \]Divide 286 by 13:
\[ 286 \div 13 = 22 \]Now, multiply by 6:
\[ 6 \times 22 = 132 \]So, the first child receives 132 pencils.
D) 132 — The calculation shows that the first child receives 132 pencils.
The question asks to identify the specific valve located between the right atrium and the right ventricle in the human heart. This requires knowledge of the anatomy of the heart and the names of its various valves.
Correct Option: D) Tricuspid valve
The tricuspid valve is situated between the right atrium and the right ventricle. It has three cusps and prevents the backflow of deoxygenated blood into the right atrium when the right ventricle contracts.
The question asks which reproductive process allows for greater variation. We need to evaluate each option based on its mechanism of generating genetic diversity.
C) Sexual reproduction allows for greater variation to be generated because it involves the fusion of gametes from two parents, genetic recombination through meiosis (crossing over and independent assortment), and the random combination of parental genes, all of which contribute to significant genetic diversity in the offspring.
A 10 kg object falls a distance of 4 m. The work done by the force of gravity acting on it is (take g = 10 m/s2)
The problem asks us to calculate the work done by the force of gravity on an object falling a certain distance. Work done by a constant force is calculated as the product of the force and the displacement in the direction of the force. In this case, the force is gravity, and the displacement is vertical.
The force of gravity (weight) is given by the formula:
\[ F = m \times g \]Work done (\(W\)) by a force is given by the formula:
\[ W = F \times h \]D) 400 J — The work done by the force of gravity is the product of the gravitational force (weight) and the vertical distance fallen. Given the mass of 10 kg and \(g = 10 \text{ m/s}^2\), the force of gravity is \(10 \text{ kg} \times 10 \text{ m/s}^2 = 100 \text{ N}\). The distance fallen is 4 m. Therefore, the work done is \(100 \text{ N} \times 4 \text{ m} = 400 \text{ J}\).
The question asks to identify the musical instrument Pandit Vishwa Mohan Bhatt is famous for playing, especially given his Grammy Award recognition. This requires knowledge of prominent Indian musicians and their associated instruments.
A) Slide Guitar player — Pandit Vishwa Mohan Bhatt is internationally acclaimed for his mastery of the Mohan Veena, which is a modified slide guitar. He even won a Grammy Award for his work with this instrument.
The question asks to evaluate the strength of arguments against displaying a health warning when an actor consumes alcohol in movies. A strong argument should be objective, universally acceptable, and directly relevant to the core issue of public health messaging versus cinematic experience. Arguments based purely on personal preference or minor inconvenience are generally considered weak.
Correct Option: C) Both I and II are not strong
When is the National Civil Services Day celebrated in India?
The question asks for the date on which National Civil Services Day is celebrated in India. This is a factual question related to important days and dates in India, which is a common topic in competitive examinations.
Correct Option: B) April 21st
The problem requires us to decode a word based on a given coding pattern. We need to identify the rule applied to transform "CARRY" into "XZIIB" and then apply the same rule to "BRING".
Word: C A R R Y
Code: X Z I I B
The pattern is that each letter in the original word is replaced by its corresponding letter from the opposite end of the alphabet. For example, A becomes Z, B becomes Y, C becomes X, and so on.
So, BRING becomes YIRMT.
D) YIRMT is the correct code for BRING, following the established pattern of replacing each letter with its reverse alphabetical counterpart.