Answer:
<em>The actual distance between the two towns is 105 km</em>
Step-by-step explanation:
<em>Lets use a proportion for this scenario</em>
<em>1/30 = 3.5/?</em>
<em>We see that the numerator is multiplied by 3.5, from 1 to 3.5</em>
<em>So we do the same to the denominator</em>
30 x 3.5 is 105 km
So your answer is 105 km
Answer:
52 because in 1 cup equivalent to 8 ounces.. if she drinks 6 cups that is 48 ounces plus the half cups 4 ounces so 52 ounces all in all
To obtain the slope of the tangent line to f(x) at x=2, we need to find f'(x) as shown below
![f^{\prime}(x)=2x-2](https://tex.z-dn.net/?f=f%5E%7B%5Cprime%7D%28x%29%3D2x-2)
Then, the slope of the tangent line we are looking for is
![f^{\prime}(2)=2\cdot2-2=2=\frac{2}{1}](https://tex.z-dn.net/?f=f%5E%7B%5Cprime%7D%282%29%3D2%5Ccdot2-2%3D2%3D%5Cfrac%7B2%7D%7B1%7D)
Remember that the slope of a line can be interpreted as shown below
![m=\frac{a}{b}\to\text{move a units on the y-ax is direction per each b units on x-axis direction}](https://tex.z-dn.net/?f=m%3D%5Cfrac%7Ba%7D%7Bb%7D%5Cto%5Ctext%7Bmove%20a%20units%20on%20the%20y-ax%20is%20direction%20per%20each%20b%20units%20on%20x-axis%20direction%7D)
Therefore, we can estimate the value of f(x) using the slope and f(2).
a) x=2.4
![f(2.4)=3.96](https://tex.z-dn.net/?f=f%282.4%29%3D3.96)
And the estimation using f(2) and f'(2) is
![\begin{gathered} f(2.4)\approx f(2)+f^{\prime}(2)\cdot(2.4-2)=3+2(0.4)=3.8 \\ \end{gathered}](https://tex.z-dn.net/?f=%5Cbegin%7Bgathered%7D%20f%282.4%29%5Capprox%20f%282%29%2Bf%5E%7B%5Cprime%7D%282%29%5Ccdot%282.4-2%29%3D3%2B2%280.4%29%3D3.8%20%5C%5C%20%20%5Cend%7Bgathered%7D)
Then, the exact value at x=2.4 is f(2.4)=3.96, and the approximated value is 3.8
![3.96-3.8=0.16](https://tex.z-dn.net/?f=3.96-3.8%3D0.16%3C0.5)
We need to repeat these steps with the remaining options.
b) x=2.5
![\begin{gathered} x=2.5 \\ \Rightarrow f(2.5)=4.25 \\ \text{and} \\ f(2.5)\approx\approx f(2)+f^{\prime}(2)\cdot(2.5-2)=3+2(0.5)=4 \\ \Rightarrow4.25-4=0.25](https://tex.z-dn.net/?f=%5Cbegin%7Bgathered%7D%20x%3D2.5%20%5C%5C%20%5CRightarrow%20f%282.5%29%3D4.25%20%5C%5C%20%5Ctext%7Band%7D%20%5C%5C%20f%282.5%29%5Capprox%5Capprox%20f%282%29%2Bf%5E%7B%5Cprime%7D%282%29%5Ccdot%282.5-2%29%3D3%2B2%280.5%29%3D4%20%5C%5C%20%5CRightarrow4.25-4%3D0.25%3C0.5%20%5Cend%7Bgathered%7D)
c) x=2.6
![\begin{gathered} x=2.6 \\ \Rightarrow f(2.6)=4.56 \\ \text{and} \\ f(2.6)\approx f(2)+f^{\prime}(2)\cdot(2.6-2)=3+2(0.6)=4.2 \\ \Rightarrow4.56-4.2=0.36](https://tex.z-dn.net/?f=%5Cbegin%7Bgathered%7D%20x%3D2.6%20%5C%5C%20%5CRightarrow%20f%282.6%29%3D4.56%20%5C%5C%20%5Ctext%7Band%7D%20%5C%5C%20f%282.6%29%5Capprox%20f%282%29%2Bf%5E%7B%5Cprime%7D%282%29%5Ccdot%282.6-2%29%3D3%2B2%280.6%29%3D4.2%20%5C%5C%20%5CRightarrow4.56-4.2%3D0.36%3C0.5%20%5Cend%7Bgathered%7D)
d) x=2.7
![\begin{gathered} x=2.7 \\ \Rightarrow f(2.7)=4.89 \\ \text{and} \\ f(2.7)\approx f(2)+f^{\prime}(2)\cdot(2.7-2)=3+2(0.7)=4.4 \\ \Rightarrow4.89-4.4=0.49](https://tex.z-dn.net/?f=%5Cbegin%7Bgathered%7D%20x%3D2.7%20%5C%5C%20%5CRightarrow%20f%282.7%29%3D4.89%20%5C%5C%20%5Ctext%7Band%7D%20%5C%5C%20f%282.7%29%5Capprox%20f%282%29%2Bf%5E%7B%5Cprime%7D%282%29%5Ccdot%282.7-2%29%3D3%2B2%280.7%29%3D4.4%20%5C%5C%20%5CRightarrow4.89-4.4%3D0.49%3C0.5%20%5Cend%7Bgathered%7D)
Then, the answer is option d. x=2.7
e) x=2.8
![\begin{gathered} x=2.8 \\ \Rightarrow f(2.8)=5.24 \\ \text{and} \\ \end{gathered}](https://tex.z-dn.net/?f=%5Cbegin%7Bgathered%7D%20x%3D2.8%20%5C%5C%20%5CRightarrow%20f%282.8%29%3D5.24%20%5C%5C%20%5Ctext%7Band%7D%20%5C%5C%20%20%5Cend%7Bgathered%7D)
Consider this option:
cos π/6=√3/2;
cos π/3=1/2;
tan π/6=1/√3;
tan π/3=√3