The top three teams qualified to compete at the state competition. At the state competition, however, scoring is done differently than at the other meets. Use the following information to create a system of equations where is the points for first place, is the points for second place, and is the points for third place.
Results:
Eagles got 5 first-place, 3 second-place, and 4 third-place finishes with a team total of 67 points.
Pirates got 2 first-place, 5 second-place, and 5 third-place finishes with a team total of 56.
Devil Rays got 6 first-place, 3 second-place and 3 third-place finishes with a team total of 72.
We can’t believe the Devil Rays came from behind and won! We need to determine how many points were awarded for each place. Use Cramer’s rule or inverse method to get the answers. Explain your process.
Answers
The equations are
5x+3y+4z=672x+5y+5z=566x+3y+3z=72Lets solve them through matrix
Let
[tex]\\ \rm\Rrightarrow A=\left[\begin{array}{ccc}\rm 5&\rm 3&\rm 4\\ \rm 2&\rm 5&\rm 5\\ \rm 6&\rm 3&\rm3\end{array}\right][/tex]
[tex]\\ \rm\Rrightarrow X=\left[\begin{array}{c}\rm x\\ \rm y\\ \rm z\end{array}\right][/tex]
[tex]\\ \rm\Rrightarrow B=\left[\begin{array}{c}\rm 67 \\ \rm 56\\ \rm 72\end{array}\right][/tex]
Let solve
Determinant of A
[tex]\\ \rm\Rrightarrow |A|[/tex]
[tex]\\ \rm\Rrightarrow 5\left|\begin{array}{cc}\rm 5&\rm 5\\ \rm 3&\rm 3\end{array}\right|-3\left|\begin{array}{cc}\rm 2&\rm 5\\ \rm 6&\rm 3\end{array}\right|+4\left|\begin{array}{cc}\rm 2&\rm 5\\ \rm 6&\rm 3\end{array}\right|[/tex]
[tex]\\ \rm\Rrightarrow 5(15-15)-3(6-30)+4(6-30)[/tex]
[tex]\\ \rm\Rrightarrow 72-96[/tex]
[tex]\\ \rm\Rrightarrow -24[/tex]
Cofactor[tex]\\ \rm\Rrightarrow A_{11}=0[/tex]
[tex]\\ \rm\Rrightarrow A_{12}=24[/tex]
[tex]\\ \rm\Rrightarrow A_{13}=-24[/tex]
[tex]\\ \rm\Rrightarrow A_{21}=3[/tex]
[tex]\\ \rm\Rrightarrow A_{22}=-9[/tex]
[tex]\\ \rm\Rrightarrow A_{23}=3[/tex]
[tex]\\ \rm\Rrightarrow A_{31}=-5[/tex]
[tex]\\ \rm\Rrightarrow A_{32}=-17[/tex]
[tex]\\ \rm\Rrightarrow A_{33}=19[/tex]
Cofactor of A[tex]\\ \rm\Rrightarrow \left[\begin{array}{ccc}\rm 0&\rm 24&\rm -24\\ \rm 3&\rm -9&\rm 3\\ \rm -5&\rm -17&\rm 19\end{array}\right][/tex]
Adjacent of A[tex]\\ \rm\Rrightarrow \left[\begin{array}{ccc}\rm 0&\rm 24&\rm -24\\ \rm 3&\rm -9&\rm 3\\ \rm -5&\rm -17&\rm 19\end{array}\right]^T[/tex]
[tex]\\ \rm\Rrightarrow \left[\begin{array}{ccc}\rm 0&\rm 3&\rm -5\\ \rm 24&\rm -9&\rm -17\\ \rm -24&\rm 3&\rm 19\end{array}\right][/tex]
Inverse of A[tex]\\ \rm\Rrightarrow \dfrac{1}{|A|}\times Adj.A[/tex]
[tex]\\ \rm\Rrightarrow \dfrac{1}{-24}\left[\begin{array}{ccc}\rm 0&\rm 3&\rm -5\\ \rm 24&\rm -9&\rm -17\\ \rm -24&\rm 3&\rm 19\end{array}\right][/tex]
Now
[tex]\\ \rm\Rrightarrow AX=B[/tex]
[tex]\\ \rm\Rrightarrow X=A^{-1}B[/tex]
[tex]\\ \rm\Rrightarrow X=\dfrac{1}{-24}\left[\begin{array}{ccc}\rm 0&\rm 3&\rm -5\\ \rm 24&\rm -9&\rm -17\\ \rm -24&\rm 3&\rm 19\end{array}\right]\left[\begin{array}{c}\rm 67\\ \rm 56\\ \rm 72\end{array}\right][/tex]
[tex]\\ \rm\Rrightarrow X=\dfrac{1}{-24}\left[\begin{array}{c}\rm -192\\ \rm -120\\ \rm -72\end{array}\right][/tex]
[tex]\\ \rm\Rrightarrow X=\left[\begin{array}{c}\rm 8\\ \rm 5\\ \rm 3\end{array}\right][/tex]
So
[tex]\\ \rm\Rrightarrow \left[\begin{array}{c}\rm x\\ \rm y\\ \rm z\end{array}\right]=\left[\begin{array}{c}\rm 8\\ \rm 5\\ \rm 3\end{array}\right][/tex]
We get
x=8y=5z=3
Related Questions
You deposit # 100 in a savings account that earns 6% annual interest compounded monthly. What
the balance after 5 years.
Answers
Answer:
$134.89.
Step-by-step explanation:
Monthly interest = 0.06/12 and its for 5*12 months.
Amount after 5 years = 100(1 + 0.06/12)^(5*12)
= 100(1 + 0.06/12)^60
= $134.89
9. Use the compound-interest formula to find the account balance A, where P is principal, r is interest rate, n is number of compounding periods per year, t is time, in years, and A is account balance.
Answers
Interest on interest, or compound interest, is the adding of interest to the principal sum of a loan or deposit. The account balance after a period of 5 years will be $1.63634×10²⁰.
What is compound interest?Interest on interest, or compound interest, is the adding of interest to the principal sum of a loan or deposit. It's the outcome of reinvesting interest rather than paying it out so that interest is received on the principal plus previously collected interest in the next quarter.
[tex]A = P(1 + \frac{r}{n})^{nt}[/tex]
A = final amount
P = initial principal balance
r = interest rate
n = number of times interest applied per time period
t = number of time periods elapsed
Given the principal amount is $12,385, the rate of interest is 7.5%, and the time period is daily. Therefore, the amount can be written as,
[tex]A = P(1 + \frac{r}{n})^{nt}\\\\A = \$12,385(1 + \frac{7.5}{365})^{(365 \times 5)}\\\\A = \$1.63634 \times 10^{20}[/tex]
Hence, the account balance after a period of 5 years will be $1.63634×10²⁰.
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If cost of 15 eggs is Rs. 75, then find out the cost of 4 dozens eggs.
a.₹ 185
b.₹ 150
c.₹ 300
d.₹ 240
Answers
Answer:
The answer is d 240. You will first find how much 1 egg costs then times 12 times 4 equal to 48 times how much 1 egg Costs equal to 240
Help me I can’t seem to figure this one out
Answers
Answer:
See below
Step-by-step explanation:
Expand it a bit to see:
= [tex]\sqrt[5]{ 2^5 * 7 * x^5 * x^5 * x * y^5 * y^3}[/tex]
=[tex]2x^2y \sqrt[5]{7xy^3}[/tex]
Find the minimum value of
\[\frac{x^2 + 9}{x}\]for $x > 0.$
I tried this but I didn't get it, could someone explain?
Answers
I hope this will help you a lot
What does this graph show
inverse
direct
neither
Answers
Answer:
i think the given graph is inverse
What percent of 4c in each expression? *2a
Answers
The percentage of 4c that's the expression 0.04c will be is 1%.
How to calculate the percentage?From the complete information, we are told to calculate the percentage of 0.04c from 4c.
This will be calculated thus:
= 0.04c/4c × 100
= 0.01
= 1%
Therefore, the percentage is 1%.
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A polygon GHIJ is inscribed in a circle. If m angle G H I equals 72 degree and m angle H I J equals 60 degree, what is m angle I J G?
A circle with an inscribed polygon G H I J.
m angle I J G equals _(blank)_ degree
Answers
The measure of angle IJG in the inscribed quadrilateral is calculated as: 110°.
What are the Opposite Angles of an Inscribed Quadrilateral?The opposite angles in a inscribed quadrilateral in a circle are said to be equal to 180 degrees, that is, they are supplementary.
angle IJG and angle GHI are opposite angles, therefore:
Angle IJG = 180 - angle GHI
Angle IJG = 180 - 70
Angle IJG = 110°
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The diameter of a circle has endpoints (13,10) and (5,16) Part A: Find the center of the circle Part B: Find the distance of the radius Part C: Write the equation of the circle.
Answers
Answer:
See below ~
Step-by-step explanation:
Part A : Center
Midpoint of diameter pointsM = (13 + 5 / 2, 10 + 16 / 2)M = (9, 13)Part B : Radius
Half of the distance between the points16 - 10 / 26/2radius = 3Part C : Equation
(x - 9)² + (y - 13)² = 3²x² - 18x + 81 + y² - 26y + 169 = 9x² - 18x + y² - 26y + 241 = 0please do all of number 2 please help (find the angle measurements that are not given
Answers
The measures of the missing angles are as follows:
a. 145°
b. 43°
c. 36°, 36° and 144°
How to find angles when lines intersect?When two lines intersect, angle relationship can be formed. The angles can be adjacent angles, vertical angles and many more.
Therefore,
a. 180 - 35 = 145 degrees( angle on a straight line)
b. 180- 47 - 90 = 43 degrees(angles on a straight line)
c. 144 degrees (vertical angles), 36 degrees and 36 degrees(angles on a point is 36 degrees)
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What two operations would you use to solve the equation 5x - 8=-18
-multiplication
-division
-addition
-subtraction
Answers
Answer:
subtraction
Step-by-step explanation:
5x-8+8=-18+8
5x=-10
5x/5=-10/5
x=-2
Answer:
addition and division
Step-by-step explanation:
5x - 8 = - 18 ( add 8 to both sides )
5x = - 10 ( divide both sides by 5 )
x = [tex]\frac{-10}{5}[/tex] = - 2
A traveler pulls a suitcase, as shown in the figure.
The force acting on the handle is 30 N at an angle of θ = 40°. How much work is done if the traveler moves 120 meters? Round to the nearest whole number.
2,314 Nm
2,758 Nm
3,085 Nm
4,156 Nm
&
A cart weighing 180 pounds rests on an incline at an angle of 35°. What is the required force to keep the cart at rest? Round to the thousandths place.
77.073 pounds
103.244 pounds
147.447 pounds
162.665 pounds
Answers
Answer:
2758 Nm
Step-by-step explanation
Work done usually depends on two things force applied and distance travelled due to applied force. In this current scenario, the force is being applied at an angle so we will have to find a component of force in the direction of the movement.
We usually find component using cos θ.
Here θ is 40°
Now, the modified equation becomes,
Work Done = Force * Distance * Component of force along the direction of distance
∴ Work = 30 N * 120 m * Cos 40°
⇒Work = 30 * 120 * 0.766
⇒Work = 2757.6 Nm
Rounding to the nearest whole number.
∴ The work done by force is 2758 Nm which is option B
The amount of work done will be 2758 Nm. Then the correct option is B. And the required force to keep the cart at rest will be 147.447 pounds. Then the correct option is C.
What is work done?The product of the force and displacement.
WD = F × d × cos θ
A traveler pulls a suitcase, as shown in the figure.
The force acting on the handle is 30 N at an angle of θ = 40°.
Then the amount of work done if the traveler moves 120 meters will be
WD = 30 × 120 × cos 40°
WD = 30 × 120 × 0.766
WD = 2758 Nm
Then the correct option is B.
A cart weighing 180 pounds rests on an incline at an angle of 35°.
The required force to keep the cart at rest will be
Fₓ = F × cos θ
Fₓ = 180 × cos 35°
Fₓ = 147.447 pounds
Then the correct option is C.
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Yo anyone from United States I gotta question
Answers
what's your question question question
The diameter of a circle is 34 inches. What is the circle's circumference?
Answers
Answer:
Hello!!!
34pi or 106.81
Please read the explanation... it always helps... i promise...
Step-by-step explanation:
Alrighty the circle circumference states that... c = 2r pi
we need to find the radius which is always half of the diameter.... so 34/2=17
the radius is 17... now all we have to do is substitute 17 into the circumference formula...
(2)17 pi = 34pi OR 106.81
or you can do Diameter multiplied by pi since the diameter is basically the radius multiplied by two... there are multiple ways you could approach this problem...
i hope this helps!!!!!!!
The circumference of the circle is approximately 106.96 inches.
What is a circle?A circle is a two-dimensional figure with a radius and circumference of 2pir.
The area of a circle is πr².
We have,
The circumference C of a circle with diameter d is given by the formula:
C = πd
Substituting the given value of diameter, we get:
C = π(34) = 34π
Using an approximation of π as 3.14, we can calculate the approximate circumference as:
C = 34 × 3.14 = 106.96 inches
Therefore,
The circumference of the circle is approximately 106.96 inches.
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Is 3/10 equivalent to 12/20 or 5/20?
Answers
Answer:
Neither?
Step-by-step explanation:
3/10 = 0.3
12/20 = 0.6
5/20 = 0.25
I would say that its up to you what you want to put.
Sorry if it doesn't help.
I was confused on how to write the quadratic in standard form
Answers
Answer:
3x² + 4x + 8 = 0
Step-by-step explanation:
Standard form of quadratic equation is
ax²+bx+c=0
standard form of quadratic equation 3x²+4x+7=-1 is
3x² + 4x + 7 = -1
3x² + 4x + 7 + 1 = 0
3x² + 4x + 8 = 0
Convert the improper fraction to a mixed number 17/3
Answers
Answer:
5 and 2/3
Step-by-step explanation:
17/3 = 17 - 3 = 14
1
14 - 3 = 11
2
11 - 3 = 8
3
8 - 3 = 5
4
5 - 3 = 2
5
So it’s 5 2/3
Pick the geometric term that matches the real-world object. cylinder cube sphere triangular prism rectangular prism
Answers
Answer:
cylinder
Step-by-step explanation:
A mascara tube is a cylinder
Which of the following options have the same value as 10% of 33?
Choose 3 answers:
0.1.0.33
® 10.33
©
1
10
33
0.1.33
10
100
33
3o
Answers
Answer:
1
Step-by-step explanation:
In the previous task you saw that the number of
moves to solve the Tower of Hanoi with n disks
given by the equation an -2-1.
What is the fewest number of moves required for
7 disks?
13
64
J
127
128
COMPLETE
ANSWER:127
Answers
Using the given equation, it is found that the fewest number of moves required for 7 disks is of 127.
What is the fewest number of moves needed with n disks?It is given by the following equation:
[tex]a_n = 2^n - 1[/tex]
In this problem, we have 7 disks, hence the fewest number of moves needed is given by:
[tex]a_7 = 2^7 - 1 = 127[/tex]
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Answer:
127
Step-by-step explanation:
Jim works as a manager at a local supermarket in dallas. jim worked 36 hours and his paycheck for this week was $1,620.51. how much does jim make per hour? round your answer to the nearest cent.
Answers
324
Step-by-step explanation:
roughly about that much
How many 90° turns will the minute hand make between 9:00 and 10:00?
Answers
Answer:
It will make 4 90° turns, because each 1/4 turn or 15 minutes is 90°.
This is geometry but I forgot how to do the equations
Answers
Answer:
Step-by-step explanation:
Those are the special triangles 45-45-90 and 30-60-90
6. y = 5 and x = 5sqrt(2)
7. y= 8 and x = 8sqrt(3)
8. x = 10sqrt(2)
Given the system below:
f(x) = 2x
g(x) = 2x
Which value(s) of x make f(x) = g(x) a true statement? If necessary, you may choose more than one
answer.
1
4
3
2
9
0
Answers
All real numbers
1. f(x) = 2x
2. g(x) = 2x
Substitute g(x) to the 2x in the first statement.
You'll get: f(x) = g(x)
Therefore, x ∈ all real numbers (any value of x would make it a true statement)
What ordered pair is a solution to
{2y=4x-10}
{-y=4x-1}
Answers
Step-by-Step Explanation:
Lets solve using Elimination Method :-
2y = 4x - 10
-4x = -10 - 2y
=> -4x + 2y = -10 (Eq. 1)
-y = 4x - 1
-4x = -1 + y
=> -4x - y = -1 (Eq. 2)
Subtracting Eq. 2 from Eq. 1 :-
=> 0 + 3y = -9
3y = -9
=> y = -9/3 = -3
Therefore, y = -3
Substitute value of ‘y’ in Eq. 2 :-
-4x - y = -1
-4x -(-3) = -1
-4x + 3 = -1
-4x = -1 - 3
-4x = -4
4x = 4
=> x = 4/4 = 1
Therefore, x = 1
Hence the ordered pair (1,-3) is the Solution to this System of Equations.
What is the perimeter of the parallelogram?
Answers
Step-by-step explanation:
b×h1÷2 use this formulae
Using Pythagorean theory
a^2 + b^2 = c^2
64 + 36 = c^2
100 = c^2
Take square root of both sides
C=10
Now add all four sides for perimeter
10 + 10 + 8 + 8 =36
See picture for the math
Ron made 16 out of 25 baskets from the foul line. Using this data, predict the
number of baskets he will make from the foul line if he shoots 200 times?
A 64
r B. 72
C. 92
D. 128
Answers
Answer:
D. 128
Step-by-step explanation:
16 out of 25 is a 64% ratio.
Easy way to get that is there is 4 25s in 100. So do 16x4 and that's 64.
Now there is 2 100s in 200. We want out of 100 so we half it. 64 and 100.
Knowing this the answer is D. 128.
Hope this helps. Brainliest would be appreciated.
Answer:
D. 128
Step-by-step explanation:
[tex]\frac{16}{25} =\frac{x}{200}[/tex]
cross multiply:
[tex]25x = 3200[/tex]
divide both sides by 25:
x = 128
A certain species of bird is in danger of becoming extinct. There were 1500 birds in 2000 and they are
decreasing at an annual rate of 6.5%.
a) If this trend continues, how many birds will be left by 2010?
b) How many birds would there have been in 1990?
Answers
a) If this trend continues, how many birds will be left by 2010 will be 525
b) The birds would there have been in 1990 will be 2475
What is rate of increase?Rate of increase or decrease is defined as the increment or decrement in any quantity over a period of time constantly.
Here we have the data that:
There were 1500 birds in 2000 and they are decreasing at an annual rate of 6.5%.
a) If this trend continues, how many birds will be left by 2010
{1500x6.25}/{100}=97.5
So the bird decrement in the one year will be =97.5 so it will be in the 10 years or upto 2010 will be
97.5x10=975
So the birds left till 2010 will be:
Q=1500-975=525
b)The birds would there have been in 1990 will be:-
=1500+975=2475
Hence If this trend continues, how many birds will be left by 2010 will be 525.The birds would there have been in 1990 will be 2475
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Students must purchase a ticket to attend the show. the committee wants to
have $500 remaining from ticket sales, after paying for equipment, lighting,
and costumes.
the tickets will cost $5.60 for each ticket, and the committee expects to sell
as many tickets as there are seats available.
what is the greatest amount of money, in dollars, the committee can spend
on equipment and costumes?
Answers
Answer:
The total number of seats is also needed to answer the question.
Step-by-step explanation:
For example, if there were 100 seats the committee would make $560 in all, therefore having $60 to spend on equipment and costumes.
label the quadrants IV
Answers
Answer:
Quadrant IV is always in the bottom right corner.