Answer:
![8\ gallons](https://tex.z-dn.net/?f=8%5C%20gallons)
Step-by-step explanation:
The total amount of sap that day is equal to
![\frac{1}{4}(2) +\frac{3}{8}(4)+\frac{1}{2}(2)+\frac{5}{8}(4)+\frac{3}{4}(2)+1(1)](https://tex.z-dn.net/?f=%5Cfrac%7B1%7D%7B4%7D%282%29%20%2B%5Cfrac%7B3%7D%7B8%7D%284%29%2B%5Cfrac%7B1%7D%7B2%7D%282%29%2B%5Cfrac%7B5%7D%7B8%7D%284%29%2B%5Cfrac%7B3%7D%7B4%7D%282%29%2B1%281%29)
![\frac{1}{2} +\frac{3}{2}+1+\frac{5}{2}+\frac{3}{2}+1](https://tex.z-dn.net/?f=%5Cfrac%7B1%7D%7B2%7D%20%2B%5Cfrac%7B3%7D%7B2%7D%2B1%2B%5Cfrac%7B5%7D%7B2%7D%2B%5Cfrac%7B3%7D%7B2%7D%2B1)
![(\frac{1}{2} +\frac{3}{2}+\frac{5}{2}+\frac{3}{2})+(1+1)](https://tex.z-dn.net/?f=%28%5Cfrac%7B1%7D%7B2%7D%20%2B%5Cfrac%7B3%7D%7B2%7D%2B%5Cfrac%7B5%7D%7B2%7D%2B%5Cfrac%7B3%7D%7B2%7D%29%2B%281%2B1%29)
![(\frac{12}{2})+(2)](https://tex.z-dn.net/?f=%28%5Cfrac%7B12%7D%7B2%7D%29%2B%282%29)
![6+2](https://tex.z-dn.net/?f=6%2B2)
![8\ gallons](https://tex.z-dn.net/?f=8%5C%20gallons)
Answer:
Step-by-step explanation:
The line is a vertical line with an equation of x = 1
y can be anything.
Draw (on graph paper) a dot at x = 1. Go down to - 3 and keep on going down to - 9. Then go up from x = 1. That's what this problem should give you.
Answer:
1) ![\sqrt{\frac{-9}{16}}=\pm\frac{3}{4}i](https://tex.z-dn.net/?f=%5Csqrt%7B%5Cfrac%7B-9%7D%7B16%7D%7D%3D%5Cpm%5Cfrac%7B3%7D%7B4%7Di)
2) ![\sqrt[3]{-0.000125}=-0.05](https://tex.z-dn.net/?f=%5Csqrt%5B3%5D%7B-0.000125%7D%3D-0.05)
3) ![\sqrt{(36g^6)}=6(g^3)](https://tex.z-dn.net/?f=%5Csqrt%7B%2836g%5E6%29%7D%3D6%28g%5E3%29)
4) ![\sqrt[3]{125x^{21}y^{24}}=5(x^{7})(y^{8})](https://tex.z-dn.net/?f=%5Csqrt%5B3%5D%7B125x%5E%7B21%7Dy%5E%7B24%7D%7D%3D5%28x%5E%7B7%7D%29%28y%5E%7B8%7D%29)
5) ![\sqrt{\frac{27x^4}{75y^2}}=\frac{3(x^2)}{5y}](https://tex.z-dn.net/?f=%5Csqrt%7B%5Cfrac%7B27x%5E4%7D%7B75y%5E2%7D%7D%3D%5Cfrac%7B3%28x%5E2%29%7D%7B5y%7D)
Step-by-step explanation:
To find : The following expression
1) Find all the real square roots of -9/16.
![\sqrt{\frac{-9}{16}}](https://tex.z-dn.net/?f=%5Csqrt%7B%5Cfrac%7B-9%7D%7B16%7D%7D)
There is no real roots because
is an imaginary number
Therefore, ![\sqrt{\frac{-9}{16}}=\pm\frac{3}{4}i](https://tex.z-dn.net/?f=%5Csqrt%7B%5Cfrac%7B-9%7D%7B16%7D%7D%3D%5Cpm%5Cfrac%7B3%7D%7B4%7Di)
2) Find all the real cube roots of -0.000125.
![\sqrt[3]{-0.000125}](https://tex.z-dn.net/?f=%5Csqrt%5B3%5D%7B-0.000125%7D)
There is a real roots.
Therefore, ![\sqrt[3]{-0.000125}=-0.05](https://tex.z-dn.net/?f=%5Csqrt%5B3%5D%7B-0.000125%7D%3D-0.05)
3) Simplify ![\sqrt{(36g^6)}](https://tex.z-dn.net/?f=%5Csqrt%7B%2836g%5E6%29%7D)
![=\sqrt{(36g^6)}](https://tex.z-dn.net/?f=%3D%5Csqrt%7B%2836g%5E6%29%7D)
![=\sqrt{6^2(g^3)^2}](https://tex.z-dn.net/?f=%3D%5Csqrt%7B6%5E2%28g%5E3%29%5E2%7D)
![=6(g^3)](https://tex.z-dn.net/?f=%3D6%28g%5E3%29)
Therefore, simplified form is ![\sqrt{(36g^6)}=6(g^3)](https://tex.z-dn.net/?f=%5Csqrt%7B%2836g%5E6%29%7D%3D6%28g%5E3%29)
4) Simplify ![\sqrt[3]{125x^{21}y^{24}}](https://tex.z-dn.net/?f=%5Csqrt%5B3%5D%7B125x%5E%7B21%7Dy%5E%7B24%7D%7D)
![=\sqrt[3]{125x^{21}y^{24}}](https://tex.z-dn.net/?f=%3D%5Csqrt%5B3%5D%7B125x%5E%7B21%7Dy%5E%7B24%7D%7D)
![=\sqrt[3]{5^3(x^{7})^3(y^{8})^3}](https://tex.z-dn.net/?f=%3D%5Csqrt%5B3%5D%7B5%5E3%28x%5E%7B7%7D%29%5E3%28y%5E%7B8%7D%29%5E3%7D)
![=5(x^{7})(y^{8})](https://tex.z-dn.net/?f=%3D5%28x%5E%7B7%7D%29%28y%5E%7B8%7D%29)
Therefore, simplified form is ![\sqrt[3]{125x^{21}y^{24}}=5(x^{7})(y^{8})](https://tex.z-dn.net/?f=%5Csqrt%5B3%5D%7B125x%5E%7B21%7Dy%5E%7B24%7D%7D%3D5%28x%5E%7B7%7D%29%28y%5E%7B8%7D%29)
5) Simplify ![\sqrt{\frac{27x^4}{75y^2}](https://tex.z-dn.net/?f=%5Csqrt%7B%5Cfrac%7B27x%5E4%7D%7B75y%5E2%7D)
![=\sqrt{\frac{27x^4}{75y^2}](https://tex.z-dn.net/?f=%3D%5Csqrt%7B%5Cfrac%7B27x%5E4%7D%7B75y%5E2%7D)
(Divide nr. and dr. by 3)
![=\sqrt{\frac{3^2(x^2)^2}{5^2y^2}](https://tex.z-dn.net/?f=%3D%5Csqrt%7B%5Cfrac%7B3%5E2%28x%5E2%29%5E2%7D%7B5%5E2y%5E2%7D)
![=\frac{3(x^2)}{5y}](https://tex.z-dn.net/?f=%3D%5Cfrac%7B3%28x%5E2%29%7D%7B5y%7D)
Therefore, simplified form is ![\sqrt{\frac{27x^4}{75y^2}}=\frac{3(x^2)}{5y}](https://tex.z-dn.net/?f=%5Csqrt%7B%5Cfrac%7B27x%5E4%7D%7B75y%5E2%7D%7D%3D%5Cfrac%7B3%28x%5E2%29%7D%7B5y%7D)
Answer
given,
side of rectangle, AB = CD = 32
BC = DA = 24
rectangle is rotated 90° clockwise about C.
then rotated 90° clockwise about D.
Path traveled by the point A for first rotation will be in circle with radius AC.
![D_1 = \dfrac{\theta_1}{360^0}\times \pi \times {AC}^2](https://tex.z-dn.net/?f=D_1%20%3D%20%5Cdfrac%7B%5Ctheta_1%7D%7B360%5E0%7D%5Ctimes%20%5Cpi%20%5Ctimes%20%7BAC%7D%5E2)
![AC = \sqrt{24^2+32^2}](https://tex.z-dn.net/?f=AC%20%3D%20%5Csqrt%7B24%5E2%2B32%5E2%7D)
AC = 40
θ₁ = 90°
![D_1 = \dfrac{90^0}{360^0}\times \pi \times {40}^2](https://tex.z-dn.net/?f=D_1%20%3D%20%5Cdfrac%7B90%5E0%7D%7B360%5E0%7D%5Ctimes%20%5Cpi%20%5Ctimes%20%7B40%7D%5E2)
D₁ = 1256.64
For the second rotation Point A will move in circular path with radius of AD
![D_2 = \dfrac{\theta_2}{360^0}\pi {AD}^2](https://tex.z-dn.net/?f=D_2%20%3D%20%5Cdfrac%7B%5Ctheta_2%7D%7B360%5E0%7D%5Cpi%20%7BAD%7D%5E2)
AD = 24
θ₁ = 90°
![D_2 = \dfrac{90^0}{360^0}\times \pi \times {24}^2](https://tex.z-dn.net/?f=D_2%20%3D%20%5Cdfrac%7B90%5E0%7D%7B360%5E0%7D%5Ctimes%20%5Cpi%20%5Ctimes%20%7B24%7D%5E2)
D₂ = 452.39
total path traveled by the point A
D = D₁ + D₂
D = 1256.64 + 452.39
D = 1709.03