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)
Answer <u>(assuming the equation can be written in point-slope form)</u>:
![y-2 = -\frac{3}{2}(x+6)](https://tex.z-dn.net/?f=y-2%20%3D%20-%5Cfrac%7B3%7D%7B2%7D%28x%2B6%29)
Step-by-step explanation:
When knowing a point the line crosses through and its slope, you can write an equation in point-slope form, or
.
1) First, find the slope of the line. Use the slope formula
and the x and y values of the two points given, then solve like so:
![\frac{(-1)-(2)}{(-4)-(-6)}\\= \frac{-1-2}{-4+6}\\= \frac{-3}{2}](https://tex.z-dn.net/?f=%5Cfrac%7B%28-1%29-%282%29%7D%7B%28-4%29-%28-6%29%7D%5C%5C%3D%20%5Cfrac%7B-1-2%7D%7B-4%2B6%7D%5C%5C%3D%20%5Cfrac%7B-3%7D%7B2%7D)
Thus, the slope is
.
2) Now, use point-slope form,
. Substitute the
,
, and
for real values.
The
represents the slope, so substitute
in its place. The
and
represent the x and y values of one point the line crosses through. Any of the two points will work, and I chose (-6,2) for this answer. So, substitute -6 for
![y-(2)= -\frac{3}{2}(x-(-6)\\y-2 = -\frac{3}{2}(x+6)](https://tex.z-dn.net/?f=y-%282%29%3D%20-%5Cfrac%7B3%7D%7B2%7D%28x-%28-6%29%5C%5Cy-2%20%3D%20-%5Cfrac%7B3%7D%7B2%7D%28x%2B6%29)
Answer: I think it’s great but you should make a stronger opinion and always have back up evidence
Step-by-step explanation:
Answer:
Amount of nuts did Glenn use = 44 cups
Step-by-step explanation:
Recipe = 56 cup of nuts
He used 12 cup fewer nuts than in the recipe
what amount of nuts did Glenn use?
Cups of nuts used = Recipe - 12
= 56 - 12
= 44 cups of nuts
Amount of nuts did Glenn use = 44 cups
6-5 is 1 7+9 is sixteen 1 divided by 16 then multiplied by sixteen is one plus nine is ten minus eight is 2 so the answer is 2