A patient takes a medication with a half life. Initially, there are 20 milligrams of the medication in the patient's system. Aft
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9514 1404 393
Answer:
(2.27, 0.96)
Step-by-step explanation:
The orthocenter is the point of intersection of altitudes of the triangle. The equation for an altitude is the equation of a line through a vertex that is perpendicular to the opposite side.
For example, the line perpendicular to side AC can be found as ...
(∆x, ∆y) = C-A = (3, -1) -(-4, 2) = (7, -3)
∆x(x -h) +∆y(y -k) = 0 . . . . perpendicular through point (h, k)
For side AC, we want the point to be B(4, 5), so the equation is ...
7(x -4) -3(y -5) = 0
7x -3y -13 = 0
Similarly, the altitude to side AB can be written as ...
(∆x, ∆y) = B -A = (4, 5) -(-4, 2) = (8, 3)
8(x -3) +3(y +1) = 0
8x +3y -21 = 0
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By the "cross multiplication method", the solution to these equations is ...
x = (-3(-21) -(3(-13))/(7(3) -8(-3)) = (63+39)/(21+24) = 102/45 = 2 4/15 ≈ 2.27
y = (-13(8) -(-21)(7))/45 = 43/45 ≈ 0.96
The orthocenter is near (2.27, 0.96).
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A graphing application confirms this result.
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<em>Additional information about the cross multiplication method</em>
For the general form equations ...
- ax +by +c = 0
- dx +ey +g = 0
The "cross multiplication method" has you write the array ...
and form the "cross products" in groups of four coefficients:
D = ae -db, X = bg -ec, Y = cd -ga
Then the solution to the set of equations is ...
1/D = x/X = y/Y ⇒ x = X/D, y = Y/D
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<em>Additional note</em>
Videos of this method show you writing the array as bcab/egde and using the final equation x/X=y/Y=1/D. The above gets the same result in a more straightforward manner.
9514 1404 393
Answer:
(x, y, z) = (-3, -1, 3)
Step-by-step explanation:
Many graphing calculators can solve matrix equations handily. Here, we use a combination of methods.
Use the last equation to write an expression for z.
z = 4 -x +4y
Substitute this into the second equation:
y -4(4 -x +4y) = -13
y -16 +4x -16y = -13
4x -15y -3 = 0
In genera form, the first equation can be written as ...
3x +y +10 = 0
Now, the solution to these two equations can be found to be ...
x = (-15(10) -1(-3))/(4(1) -3(-15)) = (-150 +3)/(4+45) = -3 . . . using "Cramer's rule"
y = -(10 +3x) = -(10 -9) = -1 . . . . from the first equation
z = 4 -(-3) +4(-1) = 3 . . . . . . . . from our equation for z
The solution to the system is (x, y, z) = (-3, -1, 3).
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<em>Additional comment</em>
Written as an augmented matrix, the system of equations is ...
Answer:
19.63%
Step-by-step explanation:
The probability of scoring 8 points is the probability of getting a dart into the green circle.
That probability is equivalent to the area of the circle divided by the total area of the square board.
The area of the circle according to , is
cm squared
The area of the whole square board is one side squared: 20 squared = 400 cm squared
The probability is therefore equal to %