Answer:
(4, - 5 )
Step-by-step explanation:
Under a counterclockwise rotation about the origin of 270°
a point (x, y ) → (- y, x ) , thus
(- 5, - 4 ) → (4, - 5 )
Explanation
We must the tangent line at x = 3 of the function:
![f(x)=(\ln x)^3.](https://tex.z-dn.net/?f=f%28x%29%3D%28%5Cln%20x%29%5E3.)
The tangent line is given by:
![y=m*(x-h)+k.](https://tex.z-dn.net/?f=y%3Dm%2A%28x-h%29%2Bk.)
Where:
• m is the slope of the tangent line of f(x) at x = h,
,
• k = f(h) is the value of the function at x = h.
In this case, we have h = 3.
1) First, we compute the derivative of f(x):
![f^{\prime}(x)=\frac{d}{dx}((\ln x)^3)=3*(\ln x)^2*\frac{d}{dx}(\ln x)=3*(\ln x)^2*\frac{1}{x}=\frac{3(\ln x)^2}{x}.](https://tex.z-dn.net/?f=f%5E%7B%5Cprime%7D%28x%29%3D%5Cfrac%7Bd%7D%7Bdx%7D%28%28%5Cln%20x%29%5E3%29%3D3%2A%28%5Cln%20x%29%5E2%2A%5Cfrac%7Bd%7D%7Bdx%7D%28%5Cln%20x%29%3D3%2A%28%5Cln%20x%29%5E2%2A%5Cfrac%7B1%7D%7Bx%7D%3D%5Cfrac%7B3%28%5Cln%20x%29%5E2%7D%7Bx%7D.)
2) By evaluating the result of f'(x) at x = h = 3, we get:
![m=f^{\prime}(3)=\frac{3}{3}*(\ln3)^2=(\ln3)^2.](https://tex.z-dn.net/?f=m%3Df%5E%7B%5Cprime%7D%283%29%3D%5Cfrac%7B3%7D%7B3%7D%2A%28%5Cln3%29%5E2%3D%28%5Cln3%29%5E2.)
3) The value of k is:
![k=f(3)=(\ln3)^3](https://tex.z-dn.net/?f=k%3Df%283%29%3D%28%5Cln3%29%5E3)
4) Replacing the values of m, h and k in the general equation of the tangent line, we get:
![y=(\ln3)^2*(x-3)+(\ln3)^3.](https://tex.z-dn.net/?f=y%3D%28%5Cln3%29%5E2%2A%28x-3%29%2B%28%5Cln3%29%5E3.)
Plotting the function f(x) and the tangent line we verify that our result is correct:
Answer
The equation of the tangent line to f(x) and x = 3 is:
Answer:
$25(h) + $130
Step-by-step explanation:
Answer:
<h3><u>Question 7</u></h3>
<u>Lateral Surface Area</u>
The bases of a triangular prism are the triangles.
Therefore, the Lateral Surface Area (L.A.) is the total surface area excluding the areas of the triangles (bases).
![\implies \sf L.A.=2(10 \times 6)+(3 \times 6)=138\:\:m^2](https://tex.z-dn.net/?f=%5Cimplies%20%5Csf%20L.A.%3D2%2810%20%5Ctimes%206%29%2B%283%20%5Ctimes%206%29%3D138%5C%3A%5C%3Am%5E2)
<u>Total Surface Area</u>
Area of the isosceles triangle:
![\implies \sf A=\dfrac{1}{2}\times base \times height=\dfrac{1}{2}\cdot3 \cdot \sqrt{10^2-1.5^2}=\dfrac{3\sqrt{391}}{4}\:m^2](https://tex.z-dn.net/?f=%5Cimplies%20%5Csf%20A%3D%5Cdfrac%7B1%7D%7B2%7D%5Ctimes%20base%20%5Ctimes%20height%3D%5Cdfrac%7B1%7D%7B2%7D%5Ccdot3%20%5Ccdot%20%5Csqrt%7B10%5E2-1.5%5E2%7D%3D%5Cdfrac%7B3%5Csqrt%7B391%7D%7D%7B4%7D%5C%3Am%5E2)
Total surface area:
![\implies \sf T.A.=2\:bases+L.A.=2\left(\dfrac{3\sqrt{391}}{4}\right)+138=167.66\:\:m^2\:(2\:d.p.)](https://tex.z-dn.net/?f=%5Cimplies%20%5Csf%20T.A.%3D2%5C%3Abases%2BL.A.%3D2%5Cleft%28%5Cdfrac%7B3%5Csqrt%7B391%7D%7D%7B4%7D%5Cright%29%2B138%3D167.66%5C%3A%5C%3Am%5E2%5C%3A%282%5C%3Ad.p.%29)
<u>Volume</u>
![\sf \implies Vol.=area\:of\:base \times height=\left(\dfrac{3\sqrt{391}}{4}\right) \times 6=88.98\:\:m^3\:(2\:d.p.)](https://tex.z-dn.net/?f=%5Csf%20%5Cimplies%20Vol.%3Darea%5C%3Aof%5C%3Abase%20%5Ctimes%20height%3D%5Cleft%28%5Cdfrac%7B3%5Csqrt%7B391%7D%7D%7B4%7D%5Cright%29%20%5Ctimes%206%3D88.98%5C%3A%5C%3Am%5E3%5C%3A%282%5C%3Ad.p.%29)
<h3><u>Question 8</u></h3>
<u>Lateral Surface Area</u>
The bases of a hexagonal prism are the pentagons.
Therefore, the Lateral Surface Area (L.A.) is the total surface area excluding the areas of the pentagons (bases).
![\implies \sf L.A.=5(5 \times 6)=150\:\:cm^2](https://tex.z-dn.net/?f=%5Cimplies%20%5Csf%20L.A.%3D5%285%20%5Ctimes%206%29%3D150%5C%3A%5C%3Acm%5E2)
<u>Total Surface Area</u>
Area of a pentagon:
![\sf A=\dfrac{1}{4}\sqrt{5(5+2\sqrt{5})}a^2](https://tex.z-dn.net/?f=%5Csf%20A%3D%5Cdfrac%7B1%7D%7B4%7D%5Csqrt%7B5%285%2B2%5Csqrt%7B5%7D%29%7Da%5E2)
where a is the side length.
Therefore:
![\implies \sf A=\dfrac{1}{4}\sqrt{5(5+2\sqrt{5})}\cdot 5^2=43.01\:\:cm^2\:(2\:d.p.)](https://tex.z-dn.net/?f=%5Cimplies%20%5Csf%20A%3D%5Cdfrac%7B1%7D%7B4%7D%5Csqrt%7B5%285%2B2%5Csqrt%7B5%7D%29%7D%5Ccdot%205%5E2%3D43.01%5C%3A%5C%3Acm%5E2%5C%3A%282%5C%3Ad.p.%29)
Total surface area:
![\sf \implies T.A.=2\:bases+L.A.=2(43.01)+150=236.02\:\:cm^2\:(2\:d.p.)](https://tex.z-dn.net/?f=%5Csf%20%5Cimplies%20T.A.%3D2%5C%3Abases%2BL.A.%3D2%2843.01%29%2B150%3D236.02%5C%3A%5C%3Acm%5E2%5C%3A%282%5C%3Ad.p.%29)
<u>Volume</u>
![\sf \implies Vol.=area\:of\:base \times height=43.011193... \times 6=258.07\:\:cm^3\:(2\:d.p.)](https://tex.z-dn.net/?f=%5Csf%20%5Cimplies%20Vol.%3Darea%5C%3Aof%5C%3Abase%20%5Ctimes%20height%3D43.011193...%20%5Ctimes%206%3D258.07%5C%3A%5C%3Acm%5E3%5C%3A%282%5C%3Ad.p.%29)