1 foot is 12 inches
12•12•12 is 1,728 cubic inches
this can help you convert because you now know how many inches are in a cubic foot, and you can divide the amount of cubic inches by 12 to find the cubic feet
2(-3n-5)-5(4n-8)
(-6n-10)+(-20n+40)
-26n-30
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)
Step-by-step explanation:
(arb)4 = 81r24
a4r4b = 81r24
a4 = 81
a = 4√81
a = 3
r4b = r24
4b = 24
b = 6
any more help just ask :)
Answer:
see explanation
Step-by-step explanation:
The equation of a line in slope- intercept form is
y = mx + c ( m is the slope and c the y- intercept )
Rearrange the given equations into this form
2x + y = 5 ( subtract 2x from both sides )
y = - 2x + 5 ← in slope- intercept form
3y = 9 - 6x ( divide all terms by 3 )
y = 3 - 2x = - 2x + 3 ← in slope- intercept form
We have
y = - 2x + 5 and y = - 2x + 3
Both equations have a slope m = - 2
The equations of lines with equal slopes are Parallel lines