Answer:
91.8°
Step-by-step explanation:
180-88.2=91.8°
Answer: 3 gallons
Step-by-step explanation:
From the question, we are informed that Connor is driving on a long road trip and currently has 9 gallons of gas in his car. We are also told that for every hour that he drives, his car uses up 2 gallons of gas.
After driving for 3 hours, his car would have used:
= 3 × 2 gallons
= 6 gallons of gas
Amount of gas left will be:
= 9 gallons - 6 gallons
= 3 gallons
Answer: Slope (m) =ΔY/ΔX=5/2=2.5
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)
Moving to the right and up, the only two points that could be placed in another point would be C or B
The answer would be 6. point C could be the image of E