Answer:
(i) ![\dfrac{df}{dL}=-\dfrac{1}{2L^2}\sqrt{\dfrac{T}{\rho}}](https://tex.z-dn.net/?f=%5Cdfrac%7Bdf%7D%7BdL%7D%3D-%5Cdfrac%7B1%7D%7B2L%5E2%7D%5Csqrt%7B%5Cdfrac%7BT%7D%7B%5Crho%7D%7D)
(ii) ![\dfrac{df}{dT}=\dfrac{1}{4L\sqrt{T\rho}}](https://tex.z-dn.net/?f=%5Cdfrac%7Bdf%7D%7BdT%7D%3D%5Cdfrac%7B1%7D%7B4L%5Csqrt%7BT%5Crho%7D%7D)
(iii) ![\dfrac{df}{d\rho}=-\dfrac{\sqrt{T}}{4L\rho^{-\frac{3}{2}}}}](https://tex.z-dn.net/?f=%5Cdfrac%7Bdf%7D%7Bd%5Crho%7D%3D-%5Cdfrac%7B%5Csqrt%7BT%7D%7D%7B4L%5Crho%5E%7B-%5Cfrac%7B3%7D%7B2%7D%7D%7D%7D)
Step-by-step explanation:
Let as consider the frequency (in Hz) of a vibrating violin string is given by
![f=\dfrac{1}{2L}\sqrt{\dfrac{T}{\rho}}](https://tex.z-dn.net/?f=f%3D%5Cdfrac%7B1%7D%7B2L%7D%5Csqrt%7B%5Cdfrac%7BT%7D%7B%5Crho%7D%7D)
(i)
Differentiate f with respect L (assuming T and rho are constants).
![\dfrac{df}{dL}=\dfrac{d}{dL}\dfrac{1}{2L}\sqrt{\dfrac{T}{\rho}}](https://tex.z-dn.net/?f=%5Cdfrac%7Bdf%7D%7BdL%7D%3D%5Cdfrac%7Bd%7D%7BdL%7D%5Cdfrac%7B1%7D%7B2L%7D%5Csqrt%7B%5Cdfrac%7BT%7D%7B%5Crho%7D%7D)
Taking out constant terms.
![\dfrac{df}{dL}=\dfrac{1}{2}\sqrt{\dfrac{T}{\rho}}\dfrac{d}{dL}\dfrac{1}{L}](https://tex.z-dn.net/?f=%5Cdfrac%7Bdf%7D%7BdL%7D%3D%5Cdfrac%7B1%7D%7B2%7D%5Csqrt%7B%5Cdfrac%7BT%7D%7B%5Crho%7D%7D%5Cdfrac%7Bd%7D%7BdL%7D%5Cdfrac%7B1%7D%7BL%7D)
![\dfrac{df}{dL}=\dfrac{1}{2}\sqrt{\dfrac{T}{\rho}}(-\dfrac{1}{L^2})](https://tex.z-dn.net/?f=%5Cdfrac%7Bdf%7D%7BdL%7D%3D%5Cdfrac%7B1%7D%7B2%7D%5Csqrt%7B%5Cdfrac%7BT%7D%7B%5Crho%7D%7D%28-%5Cdfrac%7B1%7D%7BL%5E2%7D%29)
![\dfrac{df}{dL}=-\dfrac{1}{2L^2}\sqrt{\dfrac{T}{\rho}}](https://tex.z-dn.net/?f=%5Cdfrac%7Bdf%7D%7BdL%7D%3D-%5Cdfrac%7B1%7D%7B2L%5E2%7D%5Csqrt%7B%5Cdfrac%7BT%7D%7B%5Crho%7D%7D)
(ii)
Differentiate f with respect T (assuming L and rho are constants).
![\dfrac{df}{dT}=\dfrac{d}{dT}\dfrac{1}{2L}\sqrt{\dfrac{T}{\rho}}](https://tex.z-dn.net/?f=%5Cdfrac%7Bdf%7D%7BdT%7D%3D%5Cdfrac%7Bd%7D%7BdT%7D%5Cdfrac%7B1%7D%7B2L%7D%5Csqrt%7B%5Cdfrac%7BT%7D%7B%5Crho%7D%7D)
Taking out constant terms.
![\dfrac{df}{dT}=\dfrac{1}{2L}\sqrt{\dfrac{1}{\rho}}\dfrac{d}{dT}\sqrt{T}}](https://tex.z-dn.net/?f=%5Cdfrac%7Bdf%7D%7BdT%7D%3D%5Cdfrac%7B1%7D%7B2L%7D%5Csqrt%7B%5Cdfrac%7B1%7D%7B%5Crho%7D%7D%5Cdfrac%7Bd%7D%7BdT%7D%5Csqrt%7BT%7D%7D)
![\dfrac{df}{dT}=\dfrac{1}{2L}\sqrt{\dfrac{1}{\rho}}(\dfrac{1}{2\sqrt{T}})](https://tex.z-dn.net/?f=%5Cdfrac%7Bdf%7D%7BdT%7D%3D%5Cdfrac%7B1%7D%7B2L%7D%5Csqrt%7B%5Cdfrac%7B1%7D%7B%5Crho%7D%7D%28%5Cdfrac%7B1%7D%7B2%5Csqrt%7BT%7D%7D%29)
![\dfrac{df}{dT}=\dfrac{1}{4L\sqrt{T\rho}}](https://tex.z-dn.net/?f=%5Cdfrac%7Bdf%7D%7BdT%7D%3D%5Cdfrac%7B1%7D%7B4L%5Csqrt%7BT%5Crho%7D%7D)
(iii)
Differentiate f with respect rho (assuming L and T are constants).
![\dfrac{df}{d\rho}=\dfrac{d}{d\rho}\dfrac{1}{2L}\sqrt{\dfrac{T}{\rho}}](https://tex.z-dn.net/?f=%5Cdfrac%7Bdf%7D%7Bd%5Crho%7D%3D%5Cdfrac%7Bd%7D%7Bd%5Crho%7D%5Cdfrac%7B1%7D%7B2L%7D%5Csqrt%7B%5Cdfrac%7BT%7D%7B%5Crho%7D%7D)
Taking out constant terms.
![\dfrac{df}{d\rho}=\dfrac{\sqrt{T}}{2L}\dfrac{d}{d\rho}(\rho)^{-\frac{1}{2}}}](https://tex.z-dn.net/?f=%5Cdfrac%7Bdf%7D%7Bd%5Crho%7D%3D%5Cdfrac%7B%5Csqrt%7BT%7D%7D%7B2L%7D%5Cdfrac%7Bd%7D%7Bd%5Crho%7D%28%5Crho%29%5E%7B-%5Cfrac%7B1%7D%7B2%7D%7D%7D)
![\dfrac{df}{d\rho}=\dfrac{\sqrt{T}}{2L}(-\dfrac{1}{2}(\rho)^{-\frac{3}{2}}})](https://tex.z-dn.net/?f=%5Cdfrac%7Bdf%7D%7Bd%5Crho%7D%3D%5Cdfrac%7B%5Csqrt%7BT%7D%7D%7B2L%7D%28-%5Cdfrac%7B1%7D%7B2%7D%28%5Crho%29%5E%7B-%5Cfrac%7B3%7D%7B2%7D%7D%7D%29)
![\dfrac{df}{d\rho}=-\dfrac{\sqrt{T}}{4L\rho^{-\frac{3}{2}}}}](https://tex.z-dn.net/?f=%5Cdfrac%7Bdf%7D%7Bd%5Crho%7D%3D-%5Cdfrac%7B%5Csqrt%7BT%7D%7D%7B4L%5Crho%5E%7B-%5Cfrac%7B3%7D%7B2%7D%7D%7D%7D)
The answer is B because it represents 12 x 5 which equals 60, as does 48 + 12. Hope this helps!
Answer:
No, a triangle cannot be constructed with sides of 2 in., 3 in., and 6 in.
For three line segments to be able to form any triangle you must be able to take any two sides, add their length and this sum be greater than the remaining side.
2
in.
+
3
in.
=
5
in.
5
in.
<
6
in.
For a triangle with sides 3 in., 4 in. and 5 in. which can form a triangle:
3 + 4 = 7 which is greater than 5
3 + 5 = 8 which is greater than 4
4 + 5 = 9 which is greater than 3
Step-by-step explanation:
Answer:
C. The students would prefer to play sports over going to school
Step-by-step explanation:
Hope this helps :)
can u brainlist
Answer: There are 4 students in the group.
Step-by-step explanation:
Let the number of students in the group be n
If one of them scored 9 more points , their average score would be 81 points.
So, our equation (1) will be
![\frac{\sum x+9}{n}=81\\\\\sum x+9=81n](https://tex.z-dn.net/?f=%5Cfrac%7B%5Csum%20x%2B9%7D%7Bn%7D%3D81%5C%5C%5C%5C%5Csum%20x%2B9%3D81n)
Similarly,
If one of them scored 3 points less, their average score would be 78 points.
So, our equation (2) will be
![\frac{\sum x-3}{n}=78\\\\\sum x-3=78n](https://tex.z-dn.net/?f=%5Cfrac%7B%5Csum%20x-3%7D%7Bn%7D%3D78%5C%5C%5C%5C%5Csum%20x-3%3D78n)
So, from equation (1) and (2) we get:
![81n-9=\sum x=78n+3\\\\81n-9=78n+3\\\\81n-78n=3+9\\\\3n=12\\\\n=\frac{12}{3}\\\\n=4](https://tex.z-dn.net/?f=81n-9%3D%5Csum%20x%3D78n%2B3%5C%5C%5C%5C81n-9%3D78n%2B3%5C%5C%5C%5C81n-78n%3D3%2B9%5C%5C%5C%5C3n%3D12%5C%5C%5C%5Cn%3D%5Cfrac%7B12%7D%7B3%7D%5C%5C%5C%5Cn%3D4)
Hence, there are 4 students in the group.