Answer
Verified
395.4k+ views
Hint: First, before proceeding for this, we must know the following situation to be shown in the diagram where h is the height of the rock. Then, from the figure it is clear that we can use the right angles triangle properties so that we can get the distance travelled by boat. Then, using the same property of the trigonometry for another triangle ABC. Then, we know the formula for the speed, distance and time where v is speed, t is time and d is distance as $v=\dfrac{d}{t}$, we get the final result.
Complete step-by-step answer:
In this question, we are supposed to find the time the boat will take to reach the shore.
So, before proceeding for this, we must know the following situation to be shown in the diagram where h is the height of the rock.
Now, from the figure it is clear that we can use the right angles triangle properties so that we can get the distance travelled by boat as:
$\tan {{30}^{\circ }}=\dfrac{h}{BC}$
So, by solving the above expression, we get:
$\begin{align}
& \dfrac{1}{\sqrt{3}}=\dfrac{h}{BC} \\
& \Rightarrow BC=h\sqrt{3} \\
\end{align}$
Now, by using the same property of the trigonometry for another triangle ABD is as:
$\tan {{60}^{\circ }}=\dfrac{h}{BD}$
So, by solving the above expression, we get:
$\begin{align}
& \sqrt{3}=\dfrac{h}{BD} \\
& \Rightarrow BD=\dfrac{h}{\sqrt{3}} \\
\end{align}$
Now, the distance travelled by boat in 15 minutes is given by the subtraction of the above two calculations as:
$h\sqrt{3}-\dfrac{h}{\sqrt{3}}$
Now, we know the formula for the speed, distance and time where v is speed, t is time and d is distance as:
$v=\dfrac{d}{t}$
So, by substituting the values of distance from above calculation and value of times as 15 minutes:
$\begin{align}
& v=\dfrac{h\sqrt{3}-\dfrac{h}{\sqrt{3}}}{15} \\
& \Rightarrow v=\dfrac{3h-h}{15\times \sqrt{3}} \\
& \Rightarrow v=\dfrac{2h}{15\sqrt{3}} \\
\end{align}$
So, to reach the shore, the boat needs to travel a distance of $\dfrac{h}{\sqrt{3}}$ in time t.
Then, it gives the value of time t as:
$\begin{align}
& t=\dfrac{\dfrac{h}{\sqrt{3}}}{\dfrac{2h}{15\sqrt{3}}} \\
& \Rightarrow t=\dfrac{h}{\sqrt{3}}\times \dfrac{15\sqrt{3}}{2h} \\
& \Rightarrow t=\dfrac{15}{2} \\
& \Rightarrow t=7\dfrac{1}{2} \\
\end{align}$
So, the time required by the boat to reach the shore is $7\dfrac{1}{2}$ minutes.
Hence, option (c) is correct.
Note: Now, to solve these types of the questions we need to know some of the basic trigonometric calculations of the angles from the right angles triangle. So, the basic calculation formulas are as:
$\tan \theta =\dfrac{P}{B}$ where P is perpendicular and B is base.
$\sin \theta =\dfrac{P}{H}$ where P is perpendicular and H is hypotenuse.
$\cos \theta =\dfrac{B}{H}$ where B is base and H is hypotenuse.
Complete step-by-step answer:
In this question, we are supposed to find the time the boat will take to reach the shore.
So, before proceeding for this, we must know the following situation to be shown in the diagram where h is the height of the rock.
Now, from the figure it is clear that we can use the right angles triangle properties so that we can get the distance travelled by boat as:
$\tan {{30}^{\circ }}=\dfrac{h}{BC}$
So, by solving the above expression, we get:
$\begin{align}
& \dfrac{1}{\sqrt{3}}=\dfrac{h}{BC} \\
& \Rightarrow BC=h\sqrt{3} \\
\end{align}$
Now, by using the same property of the trigonometry for another triangle ABD is as:
$\tan {{60}^{\circ }}=\dfrac{h}{BD}$
So, by solving the above expression, we get:
$\begin{align}
& \sqrt{3}=\dfrac{h}{BD} \\
& \Rightarrow BD=\dfrac{h}{\sqrt{3}} \\
\end{align}$
Now, the distance travelled by boat in 15 minutes is given by the subtraction of the above two calculations as:
$h\sqrt{3}-\dfrac{h}{\sqrt{3}}$
Now, we know the formula for the speed, distance and time where v is speed, t is time and d is distance as:
$v=\dfrac{d}{t}$
So, by substituting the values of distance from above calculation and value of times as 15 minutes:
$\begin{align}
& v=\dfrac{h\sqrt{3}-\dfrac{h}{\sqrt{3}}}{15} \\
& \Rightarrow v=\dfrac{3h-h}{15\times \sqrt{3}} \\
& \Rightarrow v=\dfrac{2h}{15\sqrt{3}} \\
\end{align}$
So, to reach the shore, the boat needs to travel a distance of $\dfrac{h}{\sqrt{3}}$ in time t.
Then, it gives the value of time t as:
$\begin{align}
& t=\dfrac{\dfrac{h}{\sqrt{3}}}{\dfrac{2h}{15\sqrt{3}}} \\
& \Rightarrow t=\dfrac{h}{\sqrt{3}}\times \dfrac{15\sqrt{3}}{2h} \\
& \Rightarrow t=\dfrac{15}{2} \\
& \Rightarrow t=7\dfrac{1}{2} \\
\end{align}$
So, the time required by the boat to reach the shore is $7\dfrac{1}{2}$ minutes.
Hence, option (c) is correct.
Note: Now, to solve these types of the questions we need to know some of the basic trigonometric calculations of the angles from the right angles triangle. So, the basic calculation formulas are as:
$\tan \theta =\dfrac{P}{B}$ where P is perpendicular and B is base.
$\sin \theta =\dfrac{P}{H}$ where P is perpendicular and H is hypotenuse.
$\cos \theta =\dfrac{B}{H}$ where B is base and H is hypotenuse.
Recently Updated Pages
Three beakers labelled as A B and C each containing 25 mL of water were taken A small amount of NaOH anhydrous CuSO4 and NaCl were added to the beakers A B and C respectively It was observed that there was an increase in the temperature of the solutions contained in beakers A and B whereas in case of beaker C the temperature of the solution falls Which one of the following statements isarecorrect i In beakers A and B exothermic process has occurred ii In beakers A and B endothermic process has occurred iii In beaker C exothermic process has occurred iv In beaker C endothermic process has occurred
The branch of science which deals with nature and natural class 10 physics CBSE
The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths
Define absolute refractive index of a medium
Find out what do the algal bloom and redtides sign class 10 biology CBSE
Prove that the function fleft x right xn is continuous class 12 maths CBSE
Trending doubts
Assertion CNG is a better fuel than petrol Reason It class 11 chemistry CBSE
How does pressure exerted by solid and a fluid differ class 8 physics CBSE
Number of valence electrons in Chlorine ion are a 16 class 11 chemistry CBSE
What are agricultural practices? Define
What does CNG stand for and why is it considered to class 10 chemistry CBSE
The rate of evaporation depends on a Surface area b class 9 chemistry CBSE
Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE
State whether the following statement is true or false class 11 physics CBSE
A night bird owl can see very well in the night but class 12 physics CBSE