Answer:
![\frac{21}{32} = x](https://tex.z-dn.net/?f=%5Cfrac%7B21%7D%7B32%7D%20%3D%20x)
Step-by-step explanation:
![\frac{x}{\frac{3}{4}} = \frac{7}{8} \\ \\ \frac{3}{4} \times \frac{7}{8} = \frac{21}{32}](https://tex.z-dn.net/?f=%5Cfrac%7Bx%7D%7B%5Cfrac%7B3%7D%7B4%7D%7D%20%3D%20%5Cfrac%7B7%7D%7B8%7D%20%5C%5C%20%5C%5C%20%5Cfrac%7B3%7D%7B4%7D%20%5Ctimes%20%5Cfrac%7B7%7D%7B8%7D%20%3D%20%5Cfrac%7B21%7D%7B32%7D)
Use multiplication [inverse of division] to figure this out.
I am joyous to assist you anytime.
Answer:
Step-by-step explanation:
From the given information:
The domain D of integration in polar coordinates can be represented by:
D = {(r,θ)| 0 ≤ r ≤ 6, 0 ≤ θ ≤ 2π) &;
The partial derivates for z = xy can be expressed as:
![y =\dfrac{\partial z}{\partial x} , x = \dfrac{\partial z}{\partial y}](https://tex.z-dn.net/?f=y%20%3D%5Cdfrac%7B%5Cpartial%20z%7D%7B%5Cpartial%20x%7D%20%2C%20x%20%3D%20%5Cdfrac%7B%5Cpartial%20z%7D%7B%5Cpartial%20y%7D)
Thus, the area of the surface is as follows:
![\iint_D \sqrt{(\dfrac{\partial z}{\partial x})^2+ (\dfrac{\partial z}{\partial y})^2 +1 }\ dA = \iint_D \sqrt{(y)^2+(x)^2+1 } \ dA](https://tex.z-dn.net/?f=%5Ciint_D%20%5Csqrt%7B%28%5Cdfrac%7B%5Cpartial%20z%7D%7B%5Cpartial%20x%7D%29%5E2%2B%20%28%5Cdfrac%7B%5Cpartial%20z%7D%7B%5Cpartial%20y%7D%29%5E2%20%2B1%20%7D%5C%20dA%20%3D%20%5Ciint_D%20%5Csqrt%7B%28y%29%5E2%2B%28x%29%5E2%2B1%20%7D%20%5C%20dA)
![= \iint_D \sqrt{x^2 +y^2 +1 } \ dA](https://tex.z-dn.net/?f=%3D%20%5Ciint_D%20%5Csqrt%7Bx%5E2%20%2By%5E2%20%2B1%20%7D%20%5C%20dA)
![= \int^{2 \pi}_{0} \int^{6}_{0} \ r \sqrt{r^2 +1 } \ dr \ d \theta](https://tex.z-dn.net/?f=%3D%20%5Cint%5E%7B2%20%5Cpi%7D_%7B0%7D%20%5Cint%5E%7B6%7D_%7B0%7D%20%5C%20r%20%20%5Csqrt%7Br%5E2%20%2B1%20%7D%20%5C%20dr%20%5C%20d%20%5Ctheta)
![=2 \pi \int^{6}_{0} \ r \sqrt{r^2 +1 } \ dr](https://tex.z-dn.net/?f=%3D2%20%5Cpi%20%5Cint%5E%7B6%7D_%7B0%7D%20%5C%20r%20%20%5Csqrt%7Br%5E2%20%2B1%20%7D%20%5C%20dr)
![= 2 \pi \begin {bmatrix} \dfrac{1}{3}(r^2 +1) ^{^\dfrac{3}{2}} \end {bmatrix}^6_0](https://tex.z-dn.net/?f=%3D%202%20%5Cpi%20%5Cbegin%20%7Bbmatrix%7D%20%5Cdfrac%7B1%7D%7B3%7D%28r%5E2%20%2B1%29%20%5E%7B%5E%5Cdfrac%7B3%7D%7B2%7D%7D%20%5Cend%20%7Bbmatrix%7D%5E6_0)
![= 2 \pi \times \dfrac{1}{3} \Bigg [ (37)^{3/2} - 1 \Bigg]](https://tex.z-dn.net/?f=%3D%202%20%5Cpi%20%5Ctimes%20%5Cdfrac%7B1%7D%7B3%7D%20%20%5CBigg%20%5B%20%2837%29%5E%7B3%2F2%7D%20-%201%20%5CBigg%5D)
![= \dfrac{2 \pi}{3} \Bigg [37 \sqrt{37} -1 \Bigg ]](https://tex.z-dn.net/?f=%3D%20%5Cdfrac%7B2%20%5Cpi%7D%7B3%7D%20%5CBigg%20%5B37%20%5Csqrt%7B37%7D%20-1%20%5CBigg%20%5D)
Average rate = distance / time
distance = average rate x time
Range of distance = 2.2(1) to 2.2(3) = 2.2 to 6.6
Required range is all real numbers from 2.2 to 6.6, inclusive.
Answer:
The dimensions are (7m-3) by (7m-3)
Step-by-step explanation:
We have the area of the rectangular field to be:
![49 {m}^{2} - 42m + 9](https://tex.z-dn.net/?f=49%20%7Bm%7D%5E%7B2%7D%20%20-%2042m%20%2B%209)
We need to factorize this expression.
Let us rewrite to obtain:
![{(7m)}^{2} - 2(7 \times 3)m + {3}^{2}](https://tex.z-dn.net/?f=%20%7B%287m%29%7D%5E%7B2%7D%20%20-%202%287%20%5Ctimes%203%29m%20%2B%20%20%7B3%7D%5E%7B2%7D%20)
We can easily see that the expression is actually a perfect square trinomial.
This gives us:
![(7m - 3)^{2}](https://tex.z-dn.net/?f=%287m%20-%203%29%5E%7B2%7D%20)
Or
![(7m - 3)(7m - 3)](https://tex.z-dn.net/?f=%287m%20-%203%29%287m%20-%203%29)