The first one since the given measurements/guide is Side, Angle, Side
2. Definition of perpendicular lines
3. Given
4. SD and DS are congruent
5. HL theorem
so we know the terminal point is at (9, -3), now, let's notice that's the IV Quadrant
![\bf (\stackrel{x}{9}~~,~~\stackrel{y}{-3})\impliedby \textit{let's find the \underline{hypotenuse}} \\\\\\ \textit{using the pythagorean theorem} \\\\ c^2=a^2+b^2\implies c=\sqrt{a^2+b^2} \qquad \begin{cases} c=hypotenuse\\ a=adjacent\\ b=opposite\\ \end{cases} \\\\\\ c=\sqrt{9^2+(-3)^2}\implies c=\sqrt{81+9}\implies c=\sqrt{90} \\\\[-0.35em] ~\dotfill](https://tex.z-dn.net/?f=%5Cbf%20%28%5Cstackrel%7Bx%7D%7B9%7D~~%2C~~%5Cstackrel%7By%7D%7B-3%7D%29%5Cimpliedby%20%5Ctextit%7Blet%27s%20find%20the%20%5Cunderline%7Bhypotenuse%7D%7D%20%5C%5C%5C%5C%5C%5C%20%5Ctextit%7Busing%20the%20pythagorean%20theorem%7D%20%5C%5C%5C%5C%20c%5E2%3Da%5E2%2Bb%5E2%5Cimplies%20c%3D%5Csqrt%7Ba%5E2%2Bb%5E2%7D%20%5Cqquad%20%5Cbegin%7Bcases%7D%20c%3Dhypotenuse%5C%5C%20a%3Dadjacent%5C%5C%20b%3Dopposite%5C%5C%20%5Cend%7Bcases%7D%20%5C%5C%5C%5C%5C%5C%20c%3D%5Csqrt%7B9%5E2%2B%28-3%29%5E2%7D%5Cimplies%20c%3D%5Csqrt%7B81%2B9%7D%5Cimplies%20c%3D%5Csqrt%7B90%7D%20%5C%5C%5C%5C%5B-0.35em%5D%20~%5Cdotfill)
![\bf cos(\theta )=\cfrac{\stackrel{adjacent}{9}}{\stackrel{hypotenuse}{\sqrt{90}}}\implies \stackrel{\textit{rationalizing the denominator}}{\cfrac{9}{\sqrt{90}}\cdot \cfrac{\sqrt{90}}{\sqrt{90}}\implies \cfrac{9\sqrt{90}}{90}}\implies \cfrac{\sqrt{90}}{10}\implies \cfrac{3\sqrt{10}}{10}](https://tex.z-dn.net/?f=%5Cbf%20cos%28%5Ctheta%20%29%3D%5Ccfrac%7B%5Cstackrel%7Badjacent%7D%7B9%7D%7D%7B%5Cstackrel%7Bhypotenuse%7D%7B%5Csqrt%7B90%7D%7D%7D%5Cimplies%20%5Cstackrel%7B%5Ctextit%7Brationalizing%20the%20denominator%7D%7D%7B%5Ccfrac%7B9%7D%7B%5Csqrt%7B90%7D%7D%5Ccdot%20%5Ccfrac%7B%5Csqrt%7B90%7D%7D%7B%5Csqrt%7B90%7D%7D%5Cimplies%20%5Ccfrac%7B9%5Csqrt%7B90%7D%7D%7B90%7D%7D%5Cimplies%20%5Ccfrac%7B%5Csqrt%7B90%7D%7D%7B10%7D%5Cimplies%20%5Ccfrac%7B3%5Csqrt%7B10%7D%7D%7B10%7D)
Answer:
80.66
Step-by-step explanation:
![L\left( x \right) = f\left( a \right) + f'\left( a \right)\left( {x - a} \right)](https://tex.z-dn.net/?f=%20L%5Cleft%28%20x%20%5Cright%29%20%3D%20f%5Cleft%28%20a%20%5Cright%29%20%2B%20f%27%5Cleft%28%20a%20%5Cright%29%5Cleft%28%20%7Bx%20-%20a%7D%20%5Cright%29%20)
![f(x, y, z) = x^2+y^2+z^2](https://tex.z-dn.net/?f=%20f%28x%2C%20y%2C%20z%29%20%3D%20x%5E2%2By%5E2%2Bz%5E2)
Since we have three variables,
![L(x, y,z) = f(a, b,c) + f_x (a, b,c) (x - a) + f_y (a, b,c) (y - b) +f_z(a,b,c)(z-c)](https://tex.z-dn.net/?f=%20L%28x%2C%20y%2Cz%29%20%3D%20f%28a%2C%20b%2Cc%29%20%2B%20f_x%20%28a%2C%20b%2Cc%29%20%28x%20-%20a%29%20%2B%20f_y%20%28a%2C%20b%2Cc%29%20%28y%20-%20b%29%20%2Bf_z%28a%2Cb%2Cc%29%28z-c%29)
![f(4, 4, 7) = 4^2+4^2+7^2=81](https://tex.z-dn.net/?f=%20f%284%2C%204%2C%207%29%20%3D%204%5E2%2B4%5E2%2B7%5E2%3D81)
![f_x (a, b,c)=2x=2*4=8\\f_y (a, b,c)=2y=2*4=8\\ f_z(a,b,c)=2z=2*7=14](https://tex.z-dn.net/?f=%20f_x%20%28a%2C%20b%2Cc%29%3D2x%3D2%2A4%3D8%5C%5Cf_y%20%28a%2C%20b%2Cc%29%3D2y%3D2%2A4%3D8%5C%5C%20f_z%28a%2Cb%2Cc%29%3D2z%3D2%2A7%3D14)
Therefore:
![L(x, y,z) = 81+ 8(x - 4) + 8 (y - 4) +14(z-7)](https://tex.z-dn.net/?f=%20L%28x%2C%20y%2Cz%29%20%3D%2081%2B%208%28x%20-%204%29%20%2B%208%20%28y%20-%204%29%20%2B14%28z-7%29)
Using the above:
f(4.02, 3.99, 6.97) =81+ 8(4.02 - 4) + 8 (3.99 - 4) +14(6.97-7)=80.66.
The approximation of
is 80.66000.
Answer:
1: D. 1/336
Explanation:
10 x 9 x 8 = 720
There are 720 ways for them to come in first, second and third place.
2. Probability = 1/8 x 1/7 x 1/6 = 1/336
The probability is 1/336 or 0.002976.