It looks like the system is
[tex]x' = \begin{bmatrix} -1 & 5 \\ -1 & 1 \end{bmatrix} x + \begin{bmatrix} \sin(t) \\ -2 \cos(t) \end{bmatrix}[/tex]
Compute the eigenvalues of the coefficient matrix.
[tex]\begin{vmatrix} -1 - \lambda & 5 \\ -1 & 1 - \lambda \end{vmatrix} = \lambda^2 + 4 = 0 \implies \lambda = \pm2i[/tex]
For [tex]\lambda = 2i[/tex], the corresponding eigenvector is [tex]\eta=\begin{bmatrix}\eta_1&\eta_2\end{bmatrix}^\top[/tex] such that
[tex]\begin{bmatrix} -1 - 2i & 5 \\ -1 & 1 - 2i \end{bmatrix} \begin{bmatrix} \eta_1 \\ \eta_2 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \end{bmatrix}[/tex]
Notice that the first row is 1 + 2i times the second row, so
[tex](1+2i) \eta_1 - 5\eta_2 = 0[/tex]
Let [tex]\eta_1 = 1-2i[/tex]; then [tex]\eta_2=1[/tex], so that
[tex]\begin{bmatrix} -1 & 5 \\ -1 & 1 \end{bmatrix} \begin{bmatrix} 1 - 2i \\ 1 \end{bmatrix} = 2i \begin{bmatrix} 1 - 2i \\ 1 \end{bmatrix}[/tex]
The eigenvector corresponding to [tex]\lambda=-2i[/tex] is the complex conjugate of [tex]\eta[/tex].
So, the characteristic solution to the homogeneous system is
[tex]x = C_1 e^{2it} \begin{bmatrix} 1 - 2i \\ 1 \end{bmatrix} + C_2 e^{-2it} \begin{bmatrix} 1 + 2i \\ 1 \end{bmatrix}[/tex]
The characteristic solution contains [tex]\cos(2t)[/tex] and [tex]\sin(2t)[/tex], both of which are linearly independent to [tex]\cos(t)[/tex] and [tex]\sin(t)[/tex]. So for the nonhomogeneous part, we consider the ansatz particular solution
[tex]x = \cos(t) \begin{bmatrix} a \\ b \end{bmatrix} + \sin(t) \begin{bmatrix} c \\ d \end{bmatrix}[/tex]
Differentiating this and substituting into the ODE system gives
[tex]-\sin(t) \begin{bmatrix} a \\ b \end{bmatrix} + \cos(t) \begin{bmatrix} c \\ d \end{bmatrix} = \begin{bmatrix} -1 & 5 \\ -1 & 1 \end{bmatrix} \left(\cos(t) \begin{bmatrix} a \\ b \end{bmatrix} + \sin(t) \begin{bmatrix} c \\ d \end{bmatrix}\right) + \begin{bmatrix} \sin(t) \\ -2 \cos(t) \end{bmatrix}[/tex]
[tex]\implies \begin{cases}a - 5c + d = 1 \\ b - c + d = 0 \\ 5a - b + c = 0 \\ a - b + d = -2 \end{cases} \implies a=\dfrac{11}{41}, b=\dfrac{38}{41}, c=-\dfrac{17}{41}, d=-\dfrac{55}{41}[/tex]
Then the general solution to the system is
[tex]x = C_1 e^{2it} \begin{bmatrix} 1 - 2i \\ 1 \end{bmatrix} + C_2 e^{-2it} \begin{bmatrix} 1 + 2i \\ 1 \end{bmatrix} + \dfrac1{41} \cos(t) \begin{bmatrix} 11 \\ 38 \end{bmatrix} - \dfrac1{41} \sin(t) \begin{bmatrix} 17 \\ 55 \end{bmatrix}[/tex]
A shop is having a 10% off sale. Find the sale price of a cap which normally costs $12?
Answer: $10.8
Step-by-step explanation: To do this, we need to find 90% of $12 because the cap is 10% off. We can do this by multiplying 12 by 0.9. 12 x0.9 = 10.8.
Use method of subtitution, will give brainliest, 20 pts
Answer:
x=3
y = -2
Step-by-step explanation:
2x+5y = -4
y = x-5
We want to use substitution
In the first equation, every time we see y, substitute x-5
2x + 5( x-5) = -4
Distribute
2x + 5x -25 = -4
Combine like terms
7x -25 = -4
Add 25 to each side
7x -25+25 = -5+25
7x = 21
Divide each side by 7
7x/7 = 21/7
x=3
Now we can find y
y = x-5
y = 3-5
y = -2
The answer is x = 3, y = -2 or (3, -2).
We are given that :
2x + 5y = -4y = x - 5Let us substitute the 2nd equation's value of y in the 1st equation.
2x + 5(x - 5) = -42x + 5x - 25 = -47x = 21x = 3Now, substitute for x in the 2nd equation.
y = 3 - 5y = -2Can anyone tell me how you could describe this answer to the equation?
[tex]f(x)=\frac{5x}{x-25}[/tex]
The end behavior of the given polynomial is that as x → -∞ or x → ∞, then, f(x) → 5
What is the end behavior of the Polynomial?We are given the polynomial;
f(x) = 5x/(x - 25)
Now, we want to find the limits as x → ±∞. Let us rearrange the given polynomial to get; f(x) = 5/(1 - (25/x))
Thus, applying limits we have;'
lim x → ±∞ [5/(1 - (25/x))]
From algebraic limit laws we know that;
If f(x) = k, then;
lim x → +∞ [f(x)] = k
Also, lim x → -∞ [f(x)] = k
Thus, applying limits at infinity to our polynomial gives;
lim x → ±∞ [5/(1 - (25/x))] = 5/(1 - 0) = 5
This is because lim x → ∞ for 1/x is 0.
Thus, f(x) has horizontal asymptotes at y = 5
Thus, we conclude that the end behavior is that as x → -∞ or x → ∞, then, f(x) → 5
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Please help with this question on a acellus thanks!
Using the law of sines, it is found that the measure of angle x is given as follows:
x = 69.9º.
What is the law of sines?Suppose we have a triangle in which:
The length of the side opposite to angle A is a.The length of the side opposite to angle B is b.The length of the side opposite to angle C is c.The lengths and the sine of the angles are related as follows:
sin(A)/A = sin(B)/B = sin(C)/C
Considering the given triangle, the following relation can be established:
sin(x)/11 = sin(28º)/x
Hence, applying cross multiplication:
sin(x) = 2sin(28º)
sin(x) = 0.938943126
x = arcsin(0.938943126)
x = 69.9º.
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What is the surface area of a rectangular prism that has a length of 12 mm, a width of 15 mm, and a height of 16 mm?
17. In a spelling competition, school X had 12 more participants than school Y and 18 fewer participants than school Z. The ratio of the number of boys to the number of girls in school X, Y and Z were 1:2, 1:5 and 2: 1 respectively. There were 50 more girls than boys in total from the 3 schools. How many participants from the 3 school were there in all?
There were 204 participants from the three schools
How to determine the number of participants?The given parameters are:
X = 12 + Y
X = Z - 18
Girls = Boys + 50
The ratios of the number of boys to girls in the schools are:
1 : 2
1 : 5
2 : 1
This means that:
School X: Boys = 1/3 and Girls = 2/3School Y: Boys = 1/6 and Girls = 5/6School Z: Boys = 2/3 and Girls = 1/3So, we have:
Boys = 1/3X + 1/6Y + 2/3Z
Girls = 2/3X + 5/6Y + 1/3Z
Substitute the above equations in Girls = Boys + 50
2/3X + 5/6Y + 1/3Z = 1/3X + 1/6Y + 2/3Z + 50
Evaluate the like terms
1/3X + 2/3Y - 1/3Z = 50
Multiply through by 3
X + 2Y - Z = 150
So, we have:
X = 12 + Y
X = Z - 18
X + 2Y - Z = 150
Substitute X = 12 + Y in X = Z - 18 and X + 2Y - Z = 150
12 + Y = Z - 18 ⇒ Y - Z = 30
12 + Y + 2Y - Z = 150 ⇒ 3Y - Z = 138
Subtract Y - Z = 30 from the equation 3Y - Z = 138 to eliminate Z
2Y = 108
Divide by 2
Y = 54
Substitute Y = 54 in X = 12 + Y
X = 12 + 54
X = 66
Substitute X = 66 in X = Z - 18
66 = Z - 18
Solve for Z
Z = 66 + 18
Z = 84
So, the total number of participants from the 3 schools is
Total = X + Y + Z
This gives
Total = 66 + 54 + 84
Evaluate the sum
Total = 204
Hence, there were 204 participants from the three schools
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Can someone help please?
[tex]r = \frac{15}{7} [/tex]
An airport parking lot charges a basic fee of $2 plus $1 per half-hour parked. what is the total charge from parking in the lot for 72 hours? a. $144 b. $146 c. $74 d. $72 please select the best answer from the choices provided a b c d
Using a linear function, the total charge from parking in the lot for 72 hours is of:
b. $146.
What is a linear function?A linear function is modeled by:
y = mx + b
In which:
m is the slope, which is the rate of change, that is, by how much y changes when x changes by 1.b is the y-intercept, which is the value of y when x = 0, and can also be interpreted as the initial value of the function.Both the basic fee, which is the y-intercept, and the hourly fee, which is the slope, are of $2, hence the cost of parking x hours is given by:
C(x) = 2x + 2
Hence the cost for parking 72 hours is:
C(72) = 2 x 72 + 2 = 2 x 73 = $146.
Which means that option b is correct.
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determine the quotient of 2/3 divided by 4/5
Which of the following relationships will have a positive slope? Select all that apply.
For this item, a non-integer answer should be entered as a fraction using / as the fraction bar.
Simplify the numerical expression.
The expression has a value equal to
The numerical expression, 2/3 ÷ 2⁴ + (3/4 + 1/6) ÷ 1/3 = 67/24 on simplification using the BODMAS rule.
In the question, we are asked to simplify the numerical expression:
2/3 ÷ 2⁴ + (3/4 + 1/6) ÷ 1/3.
To simplify the expression, we will follow the BODMAS rule, where B means Brackets, O means Of, D means Divide, M means Multiplication, A means Addition, and S means Subtraction.
2/3 ÷ 2⁴ + (3/4 + 1/6) ÷ 1/3
= 2/3 ÷ 16 + (3/4 + 1/6) ÷ 1/3 {Solving 2⁴ = 16, before proceeding BODMAS}.
= 2/3 ÷ 16 + ((9+2)/12) ÷ 1/3 {Solving Brackets by taking LCM}
= 2/3 ÷ 16 + 11/12 ÷ 1/3 {Simplifying}
= 2/3 * 1/16 + 11/12 * 3/1 {Solving divisions by taking reciprocals}
= 1/24 + 11/4 {Multiplying}
= (1 + 66)/24 {Adding using LCM}
= 67/24 {Simplifying}.
Thus, the numerical expression, 2/3 ÷ 2⁴ + (3/4 + 1/6) ÷ 1/3 = 67/24 on simplification using the BODMAS rule.
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The provided question is incomplete. The complete question is:
"Type the correct answer in the box. Use numerals instead of words. For this item, a non-integer answer should be entered as a fraction using / as the fraction bar.
Simplify the numerical expression.
2/3 ÷ 2⁴ + (3/4 + 1/6) ÷ 1/3
The expression has a value equal to."
Clark earns $31 per hour. Calculate his weekly wage if he works 38 hours per week.
Answer:
1178 because 31 dollars an hour for 38 hours you would multiply or times 31 and 38 and get 1178 for total money earned
On the day after my birthday this year, I can truthfully say: "The day after tomorrow is Monday". On which day is my birthday?
Answer: Friday
Step-by-step explanation:
Wynn is 24 years old and has decided to reduce the number of cups of coffee he buys by 2 cups per day. One cup of coffee typically costs $2.50. Assuming 30 days in a month, if he chooses to invest this money at the end of every 6 months into an investment paying 5.50% compounded semi-annually, how much will he have when he turns 69?
The total amount he would have at 69 is $343,347.81.
What is the total amount saved?The formula that can be used to determine the future value of the deferred annuity is:
Future value = annuity factor x monthly deposit
Annuity factor = {[(1+r)^n] - 1} / r
Where:
r = interest rate = 5.5 /2 = 2.75%n = number of payments = 2 x (69 - 24) = 90Amount he would save every 6 months:
amount saved per day = $2.50 x 2 = $5Amount saved per month : $5 x 30 = $150 Amount saved every 6 months = $150 x 6 = $900Future value : 900 x {[(1.0275^90) - 1] / 0.275}= $343,347.81
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What is the product of –3 and 5?
N1=731pˆ1=0. 33 n2=644pˆ2=0. 28 use this data to find the 90onfidence interval for the true difference between the population proportions
The 90% confidence interval for the true difference between the population proportions is [ - 0.0908, - 0.0092]
Given,
[tex]N_{1}[/tex] = 731
[tex]N_{2}[/tex] = 644
[tex]P_{1}[/tex] = 0.33
[tex]P_{2}[/tex] = 0.28
z score for 90% confidence interval = 1.645
Here, the confidence interval formula :
( [tex]P_{1} -P_{2}[/tex]) ± z [tex]\sqrt{\frac{P_{1}(1-P_{1}) }{N_{1} }+ \frac{P_{2}(1-P_{2}) }{N_{2} } }[/tex]
Substituting the values, we get
(0.33 - 0.28) ± 1.645 [tex]\sqrt{\frac{0.33(1-0.33)}{731}+\frac{0.28(1-0.28)}{644} }[/tex]
= 0.05 ± 1.645 [tex]\sqrt{0.0003024624+0.0003130435}[/tex]
= 0.05 ± 1.645 [tex]\sqrt{0.0006155059}[/tex]
= 0.05 ± 1.645 × 0.0248093914
= 0. 05 ± 0.0408114489
Confidence interval:
- 0.05 - 0.0408114489 = - 0.0908114489 ≈ - 0.0908
- 0.05 + 0.0408114489 = - 0.0091885511 ≈ - 0.0092
The confidence interval for the true difference between the population proportions is [ - 0.0908, - 0.0092]
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A cell phone company wants to determine the average amount of data that their smart phone customers use each month. which group would best represent a sample of the population?
The group that would best represent a sample of the study population is: C: one thousand of their customers with a current smart phone data plan.
What is a Representative Sample of a Population?A sample that best represents a population under study is a sample that is a subset of the entire population, that is, research subjects that are drawn as samples should reflect the general characteristics of the entire population without leaving out any part of the population. Also, a good sample should enable a researcher make generalization.
A sample that is a subset of a population and reflects the characteristics of the general population would not be biased.
Given the situation above, the population of the study are smart phone customers that use dat in the cell phone company. So, a sample that would better represent this general population would be customers that currently have a smart phone plan, since the average amount of data they use is what the cell pone company wants to ascertain.
Therefore, the group that would best represent a sample of the study population is: C: one thousand of their customers with a current smart phone data plan.
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What is the period of y=−5 cos( 8 π x)+3? Give an exact value. units PLEASE MAKE IT QUICK
Answer:
Maximum/Minimum Value: (0,−2) is a local minima (1/8,8) is a local maxima
Explanation: Use the derivative to find the maximum and minimum
Answer:
Period = ¹/₄
Step-by-step explanation:
The cosine function is periodic, meaning it repeats forever.
Standard form of a cosine function:
f(x) = A cos(B(x + C)) + D
where:
A = amplitude (height from the mid-line to the peak)2π/B = period (horizontal distance between consecutive peaks)C = phase shift (horizontal shift - positive is to the left)D = vertical shiftGiven function:
[tex]y=-5 \cos (8 \pi x)+3[/tex]
Comparing the given function with the standard form:
A = 5B = 8πC = 0D = 3[tex]\implies \sf Period=\dfrac{2 \pi}{B}=\dfrac{2 \pi}{8 \pi}=\dfrac{1}{4}[/tex]
Therefore, the period of the given function is ¹/₄.
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Given the function w(x) = 9x + 8, evaluate w(5). Explain all steps.
The value of the function w(5) is 53
How to evaluate the function?The function is given as:
w(x) = 9x + 8
The expression w(5) implies that the value of x is 5
i.e. x = 5
Substitute the known values in the above equation
w(5) = 9 * 5 + 8
Evaluate the product
w(5) = 45 + 8
Evaluate the sum
w(5) = 53
Hence, the value of the function w(5) is 53
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This pattern follows the rule add 14. What other features do you observe?
13, 27, 41, 55
please help meeeeee
The above pattern follows an arithmetic sequence with a common difference of 14 and first term of 13
How to solve a pattern?The pattern can be represented as follows:
13, 27, 41, 55
The first term of the sequence is 13 and the common difference is 14.
The above pattern is an arithmetic sequence.
Therefore, the following can be used to find the nth term.
nth term = a + (n - 1)d
where
n = number of termd = common differencea = first termTherefore,
a = 13
d = 27 - 13 = 14
Let's find the next term
5th term = 13 + (5 - 1)14
5th term = 13 +(4)14
5th term = 13 + 56
5th term = 69
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What is the sum of the geometric sequence -3, 18, -108,
if there are 8 terms?
Step-by-step explanation:
using gp formula
tn=ar^n-1
which our a which is the first term is = -3
our r which is t2/t1=-6
t8=(-3)(-6)^8-1
=(-3)(-6)^7
=(-3)(-279936)
=839808
Samantha and mia each left julia’s house at the same time. mia walked north at 7 kilometers per hour. samantha ran west at 11 kilometers per hour. how far apart were they after one hour? round the answer to the nearest tenth. a right triangle. a point at the angle with measure 90 degrees is labeled julia's house. a line drawn north is labeled 7 kilometers and a line drawn west is labeled 11 kilometers.
Samantha and Mia are 13 km apart from each other.
What is Pythagorean ?A right triangle's squared sides add up to the hypotenuse's squared length, according to the Pythagorean Theorem.
Mia is 7 kilometers north of Julia's home as she walked at a speed of 7 kilometers per hour.
Samantha walked at an average speed of 11 km/h, and she is now 11 kilometers west of Julia's home.
A right-angled triangle is formed by Julia's house, Mia and Samantha's locations after one hour, and the figure's points.
Hypotenuse of the triangle, which measures distance between the females
the Pythagorean theorem,
H² = P² + B²
H² = 7² + 11²
H² = 49 + 121
H = √170
H = 13.038
H = 13km
Between them, there is a 13 kilometer distance.
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If the 6am temperature in Durango, Colorado was -8° F and the temperature rose 15° by 2pm, then what would the temperature be at 2pm?
5 yd
5 yd
8 yd
5 yd
14 yd
3 yd
9 yd
6 yd
Pleasee help rn
27. Write down the co-ordinates of the points where the line y = 4x - 3 cuts the x-axis
Step-by-step explanation:
it is a line, so that is 1 point, where it cuts the x-axis.
in general that means y = 0.
so,
0 = 4x - 3
4x = 3
x = 3/4
the point, where the line cuts the x-axis is
(3/4, 0)
the weights of cars passing over a bridge have a mean of 3550 pounds and standard deviation of 870 pounds. assume that the weights of the cars passing over the bridge are normally distributed. use a calculator to find the approximate probability that the weight of a randomly selected car passing over a bridge is between 2800 and 4500
Answer:
Using the usual notations and formulas,
Using the usual notations and formulas,mean, mu = 3550
Using the usual notations and formulas,mean, mu = 3550standard deviation, sigma = 870
Using the usual notations and formulas,mean, mu = 3550standard deviation, sigma = 870Observed value, X = 3000
Using the usual notations and formulas,mean, mu = 3550standard deviation, sigma = 870Observed value, X = 3000We calculate
Using the usual notations and formulas,mean, mu = 3550standard deviation, sigma = 870Observed value, X = 3000We calculateZ= (X-mu)/sigma = (3000-3550)/870 = -0.6321839
Using the usual notations and formulas,mean, mu = 3550standard deviation, sigma = 870Observed value, X = 3000We calculateZ= (X-mu)/sigma = (3000-3550)/870 = -0.6321839Probability of weight below 3000 lbs
Using the usual notations and formulas,mean, mu = 3550standard deviation, sigma = 870Observed value, X = 3000We calculateZ= (X-mu)/sigma = (3000-3550)/870 = -0.6321839Probability of weight below 3000 lbs=P(X < 3000) = P(z < Z) = P(z < - 0.6321839) =
Using the usual notations and formulas,mean, mu = 3550standard deviation, sigma = 870Observed value, X = 3000We calculateZ= (X-mu)/sigma = (3000-3550)/870 = -0.6321839Probability of weight below 3000 lbs=P(X < 3000) = P(z < Z) = P(z < - 0.6321839) =0.2636334
Using the usual notations and formulas,mean, mu = 3550standard deviation, sigma = 870Observed value, X = 3000We calculateZ= (X-mu)/sigma = (3000-3550)/870 = -0.6321839Probability of weight below 3000 lbs=P(X < 3000) = P(z < Z) = P(z < - 0.6321839) =0.2636334Answer:
Using the usual notations and formulas,mean, mu = 3550standard deviation, sigma = 870Observed value, X = 3000We calculateZ= (X-mu)/sigma = (3000-3550)/870 = -0.6321839Probability of weight below 3000 lbs=P(X < 3000) = P(z < Z) = P(z < - 0.6321839) =0.2636334Answer:Probability that a car randomly selected is less than 3000
Using the usual notations and formulas,mean, mu = 3550standard deviation, sigma = 870Observed value, X = 3000We calculateZ= (X-mu)/sigma = (3000-3550)/870 = -0.6321839Probability of weight below 3000 lbs=P(X < 3000) = P(z < Z) = P(z < - 0.6321839) =0.2636334Answer:Probability that a car randomly selected is less than 3000=P(X < 3000) = 0.2636 (to 4 decimals)
Using the usual notations and formulas,mean, mu = 3550standard deviation, sigma = 870Observed value, X = 3000We calculateZ= (X-mu)/sigma = (3000-3550)/870 = -0.6321839Probability of weight below 3000 lbs=P(X < 3000) = P(z < Z) = P(z < - 0.6321839) =0.2636334Answer:Probability that a car randomly selected is less than 3000=P(X < 3000) = 0.2636 (to 4 decimals)Probability that a car randomly selected is greater than 3000
Using the usual notations and formulas,mean, mu = 3550standard deviation, sigma = 870Observed value, X = 3000We calculateZ= (X-mu)/sigma = (3000-3550)/870 = -0.6321839Probability of weight below 3000 lbs=P(X < 3000) = P(z < Z) = P(z < - 0.6321839) =0.2636334Answer:Probability that a car randomly selected is less than 3000=P(X < 3000) = 0.2636 (to 4 decimals)Probability that a car randomly selected is greater than 3000=1 - P(X < 3000) = 1 - 0.2636 (to 4 decimals) =0.7364 (to 4 decimals)
Is the following number rational or irrational [tex]Is the following number rational or irrational \sqrt{45\\}[/tex]
Which function is the inverse of f(x) = 2x + 3?
05²¹(x) = -1/2 x 1/2
s^²(x) = { x = 1/
2
O f¹(x) = -2x + 3
O f¹(x)=2x+3
Answer: Option 2
Step-by-step explanation:
Let f(y)=x.
[tex]x=2y+3\\\\x-3=2y\\\\y=\frac{1}{2}x-\frac{3}{2}\\\\f^{-1}(x)=\frac{1}{2}x-\frac{3}{2}[/tex]
The inverse of the function is,
⇒ f ⁻¹(x) = 1/2 (x - 3)
What is mean by Function?A relation between a set of inputs having one output each is called a function. and an expression, rule, or law that defines a relationship between one variable (the independent variable) and another variable (the dependent variable).
Given that;
The function is,
⇒ f (x) = 2x + 3
Now,
We can find the inverse of function as;
⇒ f (x) = 2x + 3
Put y = f (x);
⇒ y = 2x + 3
Solve for x;
⇒ y - 3 = 2x
⇒ x = 1/2 (y - 3)
⇒ f ⁻¹(x) = 1/2 (x - 3)
Thus, The inverse of the function is,
⇒ f ⁻¹(x) = 1/2 (x - 3)
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Jeri finds a pile of money with at least $\$200$. If she puts $\$80$ of the pile in her left pocket, gives away $\frac{1}{3}$ of the rest of the pile, and then puts the rest in her right pocket, she'll have more money than if she instead gave away $\$200$ of the original pile and kept the rest. What are the possible values of the number of dollars in the original pile of money? (Give your answer as an interval.)
Based on the amount of money that Jerl had and then the amount that she put away, the possible values of the number of dollars in the original pile of money was ( 200, 440).
What was the amount in the original pile?The amount that was above $200 that she found can be x which means that the greater side of the inequality is:
= 200 + x
She then gave away $80:
= 120 + x
She gave away 1/3 of the money so the amount left is:
= 2/3 x (120 + x)
The lesser side of the inequality:
(x + 2) - 200 = x
So then:
((120 + x) x 2) / 3 > x
240 + 2x > 3x
240 > x
x < 240
The interval that shows the possible values of the original pile of money is therefore:
( 200, 440)
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The time allotted for the Math test for 2 3/4 hours. Emma completed the test 1/2 of an hour earlier. How long did it take Emma to complete the test?
