Answer:
-3/2
Step-by-step explanation:
Let x = the integer:
2x^2 - 18 = 9x
2x^2 - 9x - 18 = 0
Now either factor that or use the quadratic formula if you're lazy like me.
x = -3/2 and 6, so x must be 6 because it has to be an integer.
A train travels 600 kilometers in 1 hour. What is the train's velocity in meters/second?
lunch Then more students joined Jaden's
Answer:
166 2/3 meters/sec.
Step-by-step explanation:
1/1 =1 12 inches/1 foot =1 you are looking for equivalents that will cancel out the unit until you can get to meters and seconds. See work in the picture.
look at the picture
The interval where the function is increasing is (3, ∞)
Interval of a functionGiven the rational function shown below
g(x) = ∛x-3
For the function to be a positive function, the value in the square root must be positive such that;
x - 3 = 0
Add 3 to both sides
x = 0 + 3
x = 3
Hence the interval where the function is increasing is (3, ∞)
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The function g is defined below.
please help
well, when it comes to fractions or rationals, they can never have a denominator that's 0, because if that ever happens, the fraction becomes undefined, so the values of "x" or namely the domain values, that we cannot have because they make the fraction undefined are those values that make the denominator 0, we can simply get them by setting the denominator to 0 and check what's "x".
[tex]x^2-9=0\implies x^2=9\implies x=\pm\sqrt{9}\implies x=\pm 3 \\\\[-0.35em] ~\dotfill\\\\ g(x)=\cfrac{x+6}{x^2-9}\hspace{5em} x\ne \begin{cases} 3\\ -3 \end{cases}[/tex]
The probability distribution for a
random variable x is given in the table.
x
-5 -3 -2 0
Probability 17
13 .33 16
2
.11
3
.10
Find the probability that -2 < x < 2
Answer:
0.6 or 60%
Step-by-step explanation:
According to the distribution table, the percent values within the interval of [- 2, 2] are:
0.33, 0.16, 0.11Add them together to get the required answer:
0.33 + 0.16 + 0.11 = 0.6 or 60%The answer is 0.6 or 60%.
The respective probabilities that lie in the interval [-2, 2] are 0.33, 0.16, and 0.11. Therefore, the probability it lies in the interval is equal to the sum of the probabilities.
0.33 + 0.16 + 0.110.49 + 0.110.6 = 60%Rewrite the expression
(10¹) ³ =
Answer:
1000
Step-by-step explanation:
The graphs of f(x) and g(x) are shown below.
f(x) = -x g(x) =2x
Which of the following is the graph of (g-f)(x)?
[tex](g-f)(x)=g(x)-f(x)=2x-(-x)=3x[/tex]
The graph is shown in the attached image.
Rotate the given triangle 270° counter-clockwise about the origin.
[tex]\left[\begin{array}{ccc}-1&2&2\\-1&-1&3\end{array}\right][/tex]
[tex]\left[\begin{array}{ccc}-1&[?]&[?]\\1&[?]&[?]\end{array}\right][/tex]
The rotation of the triangle [tex]\left[\begin{array}{ccc}-1&2&2\\-1&-1&3\end{array}\right][/tex] 270° counter-clockwise about the origin gives [tex]\left[\begin{array}{ccc}-1&-1&3\\1&-2&-2\end{array}\right][/tex]
What is transformation?Transformation is the movement of a point from its initial location to a new location. Types of transformation are rotation, reflection, translation and dilation.
If a point (x, y) is rotated 270° counter-clockwise about the origin, the new point is (y, -x)
The rotation of the triangle [tex]\left[\begin{array}{ccc}-1&2&2\\-1&-1&3\end{array}\right][/tex] 270° counter-clockwise about the origin gives [tex]\left[\begin{array}{ccc}-1&-1&3\\1&-2&-2\end{array}\right][/tex]
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the center of a circle is on the line y=2x and the line x=1 is tangent to the circle at (1,6).find the center and the radius
The radius and the center of the circle are 4 units and (1,2), respectively
How to determine the center and the radius?The center of the circle is on
y = 2x and x = 1
Substitute x = 1 in y = 2x
y = 2 * 1
Evaluate
y = 2
This means that the center is
Center = (1, 2)
Also, we have the point of tangency to be:
(x, y) = (1, 6)
This point and the center have the same x-coordinate.
So, the distance between this point and the center is
d = 6 - 2
d = 4
This represents the radius
Hence, the radius and the center of the circle are 4 units and (1,2), respectively
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in ΔWXY, m∠W = (2x - 12)°, m∠X = (2x +17)°, and m∠Y = (9x - 7)°. Find m∠X.
Answer:
45°
Step-by-step explanation:
Angles in a triangle add to 180 degrees, so
2x - 12 + 2x + 17 + 9x - 7 = 18013x - 2 = 18013x = 182x = 14So, the measure of angle X is 2(14)+17=45°
What is the y-intercept of the function f(x) = -2/9x + 1/3?
Answer: (0,1/3)
Step-by-step explanation:
Answer:
The y-intercept of the function f(x) = -2/9x + 1/3 is 1/3.
Step-by-step explanation:
Given, function is
f(x) = -2/9x + 1/3.
The y-intercept is the point where the graph intersects the y-axis.
The y-intercept of a graph is (are) the point(s) where the graph intersects the y-axis.
We know that the x-coordinate of any point on the y-axis is 0.
So the x-coordinate of a y-intercept is 0.
To find the y - intercept set x = 0.
f(0) = (2/9 . 0) + 1/3
f(0) = 0 + 1/3
f(0) = 1/3.
Suppose you pay back $575 on a $525 loan you had for 75 days. What was your simple annual interest rate? State your result to the nearest hundredth of a percent.
The simple annual interest rate for the $ 525 loan is equal to 46.35 %.
What is the interest rate behind a pay back?
In this situation we assume that the loan does not accumulate interests continuously in time. Hence, the interest rate for paying the loan back 75 days later is:
575 = 525 · (1 + r/100)
50 = 525 · r /100
5000 = 525 · r
r = 9.524
The loan has an interest rate of 9.524 % for 75 days. Simple annual interest rate is determine by rule of three:
r' = 9.524 × 365/75
r' = 46.350
The simple annual interest rate for the $ 525 loan is equal to 46.35 %.
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Identify an equation in point-alope form for the line perpendicular to y-x-7 that passes through (-2,-6).
Answer:
[tex]y + 6 = -1(x + 2).[/tex]
Step-by-step explanation:
Let's find the general equation of the given line:
[tex]y - x - 7 = 0\\\\y = x + 7.[/tex]
We can see that [tex]m = 1.[/tex]
Thus, the slope of any perpendicular line to the line [tex]y = x + 7[/tex] is [tex]-1.[/tex]
Given that the perpendicular line passes through (-2, -6), its point-slope form equation is as given:
[tex]y - y_1 = m(x - x_1)\\\\y - (-6) = -1(x - (-2))\\\\y + 6 = -1(x + 2).[/tex]
Find the measure of X
Answer:
[tex]x[/tex] = 67°
Step-by-step explanation:
As we can see in the diagram, ∠[tex]x[/tex] and the 78° angle together make up the 145° angle.
∴ [tex]x[/tex] + 78° = 145°
⇒ [tex]x[/tex] = 145° - 78°
⇒ [tex]x[/tex] = 67°
Janet Lopez is establishing an investment portfolio that will include stock and bond funds. She has $720,000 to invest, and she does not want the portfolio to include more than 65% stocks. The average annual return for the stock fund she plans to invest in is 18%, whereas the average annual return for the bond fund is 6%. She further estimates that the most she could lose in the next year in the stock fund is 22%, whereas the most she could lose in the bond fund is 5%. To reduce her risk, she wants to limit her potential maximum losses to $100,000.
a. Formulate a linear programming model for this problem.
Based on the amounts that Janet Lopez has to invest in stocks and bonds, the linear programming model would be:
0.18x + 0.06y = maximized returns 0.22x + 0.05y ≤ 100,000x + y ≤ 720,000x/ y + y ≤0.65What is the linear programming model?The return on stocks (x) is 18% and the return on bonds (y) is 6%, The objective function:
= 0.18x + 0.06y
There are constraints to watch out for:
Maximum to lose on stocks is 22% and on bonds is 5% but these are to be less than the total amount of $100,000.
0.22x + 0.05y ≤ 100,000
The total amount to invest is $72,000 which means that both bonds and stocks need to be less than this amount:
x + y ≤ 720,000
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imes A
20) A clinical trial was conducted using a new method designed to increase
the probability of conceiving a girl. As of this writing, 914 babies were
born to parents using the new method, and 877 of them were girls. Use a
(b).01 significance level to test the claim that the new method is effective in
increasing the likelihood that a baby will be a girl. Use the P-value method
and the normal distribution as an approximation to the binomial
distribution.
(10)
a. Identify the null and alternative hypothesis.
b. Compute the test statistic z.ollowenien e to jaata ni(8)
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c. What is the P-value?
d. What is the conclusion about the null hypothesis?
e. What is the final conclusion?
The test statistic is 27.8484, the p-value is 0, and the final conclusion is that the null hypothesis H is rejected.
Given that 914 babies were born to parents who used the new method, and 877 of them were girls, the significance level to test the claim that the new method is effective in increasing the likelihood of a baby being a girl, is 0.01.
The following information is provided: the sample size is N = 914, the number of favorable cases is X = 877, and the sampling ratio is
pˉ = X / N
Pˉ = 877/914
pˉ = 0.9595 and the significance level is α = 0.01
(a) Hypothesis Zero and Alternative
The following null and alternative hypotheses should be tested:
null: p = 0.5
Alternative: p> 0.5
This is equivalent to a right-tailed test, which requires a z-test for a proportion of the population.
(b) Critical Value
Based on the information provided, the significance level is α = 0.01, so the critical value for this right-tailed test is Zc = 2.3263. This can be found using Excel or the Z distribution table. Region of rejection
The rejection area for this test on the right side is Z> 2.3263
Test statistics
The z-statistic is calculated as follows:
[tex]\begin{aligned}Z&=\frac{\bar{p}-p}{\sqrt{\frac{p(1-p)}{n}}}\\ &=\frac{0.9595-0.50}{\sqrt{\frac{0.5(1-0.5)}{914}}}\\ &=\frac{0.4595}{0.0165}\\ &=27.8484\end[/tex]
(c) The p-value
The p-value is the probability that the sample results are extreme or more extreme than the sample results obtained, assuming the null hypothesis is true. In this case,
the p-value is p = P (Z> 27.8484) = 0
(d) The decision on the null hypothesis
Using the traditional method
Since we observe that Z = 27.8484> Zc = 2.3263, we conclude that the null hypothesis is rejected.
Using the p-value method
Using the approximation of the P-value: the p-value is p = 0, and since p = 0≤0.01 we conclude that the null hypothesis is rejected.
(e) Conclusion
It is concluded that the null hypothesis H is rejected. Therefore, there is sufficient evidence to state that the population proportion p is greater than 0.5, at the significance level of 0.01.
Hence, for 914 babies born to parents using the new method, of which 877 were girls, the test statistic is 27.8484, the p-value is 0, and the final conclusion is that the null hypothesis is rejected.
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The price, p, for different size orders of custom printed shirts, n, is given in the table.
Number of shirts ordered (n) 1 5 20 100
Price of order (p) $35 $75 $225 $1025
Can a linear equation be used to model the situation? If it can, what is the slope and the y-intercept of the equation?
linear: slope = 40, y-intercept = 35
linear: slope = 10, y-intercept = 25
A linear equation cannot be used.
linear: slope = 15, y-intercept = 0
Yes a Linear Equation can be used and the slope and the y-intercept of the equation are respectively; B: slope = 10, y-intercept = 25
How to Write an equation in Slope Intercept Form?
We are given the coordinates;
(n, p) = (1, 35), (5, 75), (20, 225), (100, 1025)
Now, the way to find the slope is;
m = (y2 - y1)/(x2 - x1)
m = (75 - 35)/(5 - 1)
m = 10
The equation that models this is;
(y - 35)/(x - 1) = 10
y - 35 = 10x - 10
y = 10x + 25
Thus, y-intercept is at x = 0.
y = 10(0) + 25
y = 25
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A stock lost 8 1/4 points on Monday and then another 2 3/4 points on Tuesday. On Wednesday, it gained 12 points.
What was the net gain or loss of the stock for these three days?
The net gain of the stock for these three days is 1 points
Net gainMonday = - 8 1/4 pointsTuesday = -2 3/4 pointsWednesday = 12 pointsNet gain / net loss = Monday + Tuesday + Wednesday
= - 8 1/4 + (-2 3/4) + 12
= - 8 1/4 - 2 3/4 + 12
= -11 + 12
Net gain = 1 point
Therefore, the net gain of the stock for these three days is 1 points
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what is 73 divdied by 84,649
Answer:
0.000862
Hope this helps :) If you have anymore questions just comment
Answer:
0.000862384670817 that is the answer
Suppose you were offered this choice:
ONE supersized cone for $8 or FOUR regular-sized cones for $5.
The radius of the large cone is 5 inches, and its height is 12 inches.
Each of the smaller cones has a radius of 2.5 inches and a height of 6 inches.
Does the large cone hold more ice cream, or do the four smaller cones combined?
Which is the better deal? Provide mathematical justification for your answer.
The large cone can hold more ice cream than the four smaller cones combined.
The large cone is a better deal.
What is a cone?A cone is a three-dimensional structure with a round base and a point at the top, known as the vertex.
A cone is a three-dimensional solid geometric object with a point at the top and a circular base. A cone has a vertex and one face. For a cone, there are no edges.
The radius, height, and slant height of the cone are its three constituent parts.
For the supersized cone,
Volume = π(5)(5)(12)/3 = 314.29 cubic inches
For a small cone,
Volume= π(2.5)(2.5)(6)/3 = 39.29 cubic inches
Combined volume of 4 such cones = 4*39.29 = 157.16 cubic inches
As the volume of the larger cone is more, thus it holds more ice cream. Also it is a better deal, as it costs less than 4 small cones.
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suppose you are conducting a survey about the amount grocery store baggers are tipped for helping customers to their cars .for a similar simulated population with 50 respondents the population mean is $1.73 and the standard deviation is $0.657
about 68% of the sample mean fall with in the intervals $_______ and $________
about 99.7% of the sample mean fall with in the intervals of $-------- and $
Using the Empirical Rule and the Central Limit Theorem, we have that:
About 68% of the sample mean fall with in the intervals $1.64 and $1.82.About 99.7% of the sample mean fall with in the intervals $1.46 and $2.What does the Empirical Rule state?It states that, for a normally distributed random variable:
Approximately 68% of the measures are within 1 standard deviation of the mean.Approximately 95% of the measures are within 2 standard deviations of the mean.Approximately 99.7% of the measures are within 3 standard deviations of the mean.What does the Central Limit Theorem state?By the Central Limit Theorem, the sampling distribution of sample means of size n has standard deviation [tex]s = \frac{\sigma}{\sqrt{n}}[/tex].
In this problem, the standard deviation of the distribution of sample means is:
[tex]s = \frac{0.657}{\sqrt{50}} = 0.09[/tex]
68% of the means are within 1 standard deviation of the mean, hence the bounds are:
1.73 - 0.09 = $1.64.1.73 + 0.09 = $1.82.99.7% of the means are within 3 standard deviations of the mean, hence the bounds are:
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Answer:
About 68% of the sample means fall within the interval $1.64 and $1.82.
About 99.7% of the sample means fall within the interval $1.45 and $2.01.
Step-by-step explanation:
To verify the given intervals, we need to calculate the standard error of the mean (SE) for a sample size of 50 using the population mean and standard deviation provided.
The standard error of the mean (SE) can be calculated as:
SE = population standard deviation / √(sample size)
Given that the population mean is $1.73 and the population standard deviation is $0.657, and the sample size is 50:
SE = $0.657 / √50 ≈ $0.09299 (rounded to 5 decimal places)
Now, we can calculate the intervals:
For the interval where about 68% of the sample means fall:
Interval = (Mean - 1 * SE, Mean + 1 * SE)
Interval = ($1.73 - $0.09299, $1.73 + $0.09299)
Interval ≈ ($1.63701, $1.82299)
So, about 68% of the sample means fall within the interval $1.64 and $1.82, which matches the given statement.
For the interval where about 99.7% of the sample means fall:
Interval = (Mean - 3 * SE, Mean + 3 * SE)
Interval = ($1.73 - 3 * $0.09299, $1.73 + 3 * $0.09299)
Interval ≈ ($1.54703, $1.91297)
So, about 99.7% of the sample means fall within the interval $1.55 and $1.91, which is different from the given statement.
The correct interval for about 99.7% of the sample means, rounded to the nearest hundredth, is $1.55 and $1.91, not $1.45 and $2.01 as mentioned in the statement.
adjoint of [1 0 2 -1] is
The adjoint of the matrix [tex]\left[\begin{array}{cc}1&0\\2&-1\end{array}\right][/tex] is [tex]\left[\begin{array}{cc}-1&0\\-2&1\end{array}\right][/tex]
How to determine the adjoint?The matrix is given as:
[tex]\left[\begin{array}{cc}1&0\\2&-1\end{array}\right][/tex]
For a matrix A be represented as:
[tex]A = \left[\begin{array}{cc}a&b\\c&d\end{array}\right][/tex]
The adjoint is:
[tex]Adj = \left[\begin{array}{cc}d&-b\\-c&a\end{array}\right][/tex]
Using the above format, we have:
[tex]Adj = \left[\begin{array}{cc}-1&0\\-2&1\end{array}\right][/tex]
Hence, the adjoint of the matrix [tex]\left[\begin{array}{cc}1&0\\2&-1\end{array}\right][/tex] is [tex]\left[\begin{array}{cc}-1&0\\-2&1\end{array}\right][/tex]
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If a 17-foot ladder makes a 71° angle with the ground, how many feet up a wall will it reach? Round your answer to the nearest tenth.
Given the length of the ladder and the angle it made with the ground, it will reach 16.7 ft up the wall.
How many feet up the wall will the ladder reach?Given that;
Angle with the ground θ = 71°Length of the ladder / hypotenuse = 17ftLength of wall = xTo determine the length wall from the tip of the ladder to the ground level, we use trigonometric ratio since the scenario forms a right angle triangle.
Sinθ = Opposite / Hypotenuse
Sin( 71° ) = x / 17ft
x = Sin( 71° ) × 17ft
x = 0.9455 × 17ft
x = 16.1 ft
Given the length of the ladder and the angle it made with the ground, it will reach 16.7 ft up the wall.
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pls help I dont understand this
Answer:
D
Step-by-step explanation:
evaluate the sin, cos and tan for 120deg without using a calculator
Using equivalent angles, we have that the measures are given as follows:
[tex]\sin{120^\circ} = \frac{\sqrt{3}}{2}[/tex][tex]\cos{120^\circ} = -\frac{1}{2}[/tex][tex]\tan{120^\circ} = -\sqrt{3}[/tex]What are equivalent angles?Each angle on the second, third and fourth quadrants will have an equivalent on the first quadrant.
120º is in the second quadrant, hence the equivalent on the first quadrant is:
180º - 120º = 60º.
The sine on the second quadrant is positive, hence:
[tex]\sin{120^\circ} = \sin{60^{\circ}} = \frac{\sqrt{3}}{2}[/tex]
The cosine on the second quadrant is negative, hence:
[tex]\cos{120^\circ} = \cos{60^{\circ}} = -\frac{1}{2}[/tex]
The tangent is given by the sine divided by the cosine, hence:
[tex]\tan{120^\circ} = \frac{\sin{120^\circ}}{\cos{120^\circ}} = -\sqrt{3}[/tex]
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Triangles ABC and A''B''C'' are similar. Identify the type of reflection performed and the scale factor of dilation.
The type of reflection performed and the scale factor of dilation of the diagram are; Reflection over the x-axis and a scale factor dilation of 2.
How to Identify Transformation used on diagram?From the given diagram, we are told that triangle ABC is transformed into Triangle A"B"C".
Now, we are told that both triangles are similar and as such if they are similar then, it means each corresponding side has the same ratio. Thus, there was a dilation by a scale factor of 2.
Now, the reflection that took place after the dilation would be a reflection over the x-axis.
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Find the period of the function y = 2∕3 cos(4∕7x) + 2. Question 11 options: A) 7∕2π B) 4∕7π C) 3π D) 7∕4π
The period of the function y = 2∕3 cos(4∕7x) + 2. is 7/2π
How to determine the period of the function?The function is given as:
y = 2∕3 cos(4∕7x) + 2.
The above function is a cosine function
A cosine function is represented as:
y = A cos(B(x + C)) + D
Where the period is
Period = 2π/B
By comparing the equations, we have
B = 4/7
Substitute B = 4/7 in Period = 2π/B
Period = 2π/(4/7)
Express as product
Period = 2π * 7/4
Divide 4 by 2
Period = π * 7/2
Evaluate the product
Period = 7/2π
Hence, the period of the function y = 2∕3 cos(4∕7x) + 2. is 7/2π
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A bag contains 240 marbles that are either red, blue, or green. The ratio of red to blue to green marbles is 5:2:1. If one-third of the red marbles and two-thirds of the green marbles are removed, what fraction of the remaining marbles in the bag will be blue?
A. 6/17
B. 1/2
C. 6/13
D. 7/18
The fraction of the remaining marbles in the bag will be blue is 6/17.
What fraction of the remaining marbles in the bag will be blue?The first step is to determine the initial number of marbles in the bag:
Initial number of red marbles in the bag : (5/8) x 240 = 150
Initial number of blue marbles in the bag : (2/8) x 240 = 60
Initial number of green marbles in the bag : 240 - 150 - 60 = 30
Number of red marbles remaining after 1/3 is removed = (1 - 1/3) x 150
2/3 x 150 = 100
Number of green marbles remaining after 2/3 is removed = (1 - 2/3) x 30
1/3 x 30 = 10
Total number of marbles now in the bag : 100 + 10 + 60 = 170
Fraction of blue marbles = 60 / 170 = 6/17
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Please help me with this problem. Seriously desperate again !!
The dimensions are 7 inches by 17 inches.
What is the area of rectangle?
Let the length be l inches
Let the breadth be b inches
Area = l*b
We can find dimensions as shown below:
Let the length be x inches
Let the width be y inches
Area = 119 square inches
x=3+2y (1)
Area = l*w
119 = x*y
Putting value of x
119 = (3+2y) *y
119 = 3y+2y^2
2y^2+3y-119=0
2y^2-14y+17y-119=0
2y(y-7) +17(y-7) =0
(y-7) (2y+17) =0
y=7, -17/2
y cannot be negative
so, y = 7
Putting in equation (1)
x=3+2(7)
= 3+14
= 17
Hence, the dimensions are 7 inches by 17 inches.
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A tank originally contains 100 gallon of fresh water. Then water containing 0.5 Lb of salt per gallon is pourd into the tank at a rate of 2 gal/minute, and the mixture is allowed to leave at the same rate. After 10 minute the process is stopped, and fresh water is poured into the tank at a rate of 2 gal/min, with the mixture again leaving at the same rate. Find the amount of salt in the tank at end of an additional 10 minutes.
Let [tex]S(t)[/tex] denote the amount of salt (in lbs) in the tank at time [tex]t[/tex] min up to the 10th minute. The tank starts with 100 gal of fresh water, so [tex]S(0)=0[/tex].
Salt flows into the tank at a rate of
[tex]\left(0.5\dfrac{\rm lb}{\rm gal}\right) \left(2\dfrac{\rm gal}{\rm min}\right) = 1\dfrac{\rm lb}{\rm min}[/tex]
and flows out with rate
[tex]\left(\dfrac{S(t)\,\rm lb}{100\,\mathrm{gal} + \left(2\frac{\rm gal}{\rm min} - 2\frac{\rm gal}{\rm min}\right)t}\right) \left(2\dfrac{\rm gal}{\rm min}\right) = \dfrac{S(t)}{50} \dfrac{\rm lb}{\rm min}[/tex]
Then the net rate of change in the salt content of the mixture is governed by the linear differential equation
[tex]\dfrac{dS}{dt} = 1 - \dfrac S{50}[/tex]
Solving with an integrating factor, we have
[tex]\dfrac{dS}{dt} + \dfrac S{50} = 1[/tex]
[tex]\dfrac{dS}{dt} e^{t/50}+ \dfrac1{50}Se^{t/50} = e^{t/50}[/tex]
[tex]\dfrac{d}{dt} \left(S e^{t/50}\right) = e^{t/50}[/tex]
By the fundamental theorem of calculus, integrating both sides yields
[tex]\displaystyle S e^{t/50} = Se^{t/50}\bigg|_{t=0} + \int_0^t e^{u/50}\, du[/tex]
[tex]S e^{t/50} = S(0) + 50(e^{t/50} - 1)[/tex]
[tex]S = 50 - 50e^{-t/50}[/tex]
After 10 min, the tank contains
[tex]S(10) = 50 - 50e^{-10/50} = 50 \dfrac{e^{1/5}-1}{e^{1/5}} \approx 9.063 \,\rm lb[/tex]
of salt.
Now let [tex]\hat S(t)[/tex] denote the amount of salt in the tank at time [tex]t[/tex] min after the first 10 minutes have elapsed, with initial value [tex]\hat S(0)=S(10)[/tex].
Fresh water is poured into the tank, so there is no salt inflow. The salt that remains in the tank flows out at a rate of
[tex]\left(\dfrac{\hat S(t)\,\rm lb}{100\,\mathrm{gal}+\left(2\frac{\rm gal}{\rm min}-2\frac{\rm gal}{\rm min}\right)t}\right) \left(2\dfrac{\rm gal}{\rm min}\right) = \dfrac{\hat S(t)}{50} \dfrac{\rm lb}{\rm min}[/tex]
so that [tex]\hat S[/tex] is given by the differential equation
[tex]\dfrac{d\hat S}{dt} = -\dfrac{\hat S}{50}[/tex]
We solve this equation in exactly the same way.
[tex]\dfrac{d\hat S}{dt} + \dfrac{\hat S}{50} = 0[/tex]
[tex]\dfrac{d\hat S}{dt} e^{t/50} + \dfrac1{50}\hat S e^{t/50} = 0[/tex]
[tex]\dfrac{d}{dt} \left(\hat S e^{t/50}\right) = 0[/tex]
[tex]\hat S e^{t/50} = \hat S(0)[/tex]
[tex]\hat S = 50 \dfrac{e^{1/5}-1}{e^{1/5}} e^{-t/50}[/tex]
After another 10 min, the tank has
[tex]\hat S(10) = 50 \dfrac{e^{1/5}-1}{e^{1/5}} e^{-1/5} = 50 \dfrac{e^{1/5}-1}{e^{2/5}} \approx \boxed{7.421}[/tex]
lb of salt.
Point S is on line segment \overline{RT}
RT
. Given ST=2x,ST=2x, RT=4x,RT=4x, and RS=4x-4,RS=4x−4, determine the numerical length of \overline{RS}.
RS
.
[tex]\underset{\leftarrow \qquad \textit{\LARGE 4x}\qquad \to }{R\stackrel{4x-4}{\rule[0.35em]{7em}{0.25pt}} S\stackrel{2x}{\rule[0.35em]{20em}{0.25pt}}T} \\\\\\ RT~~ - ~~ST~~ = ~~RS\implies \stackrel{RT}{4x} - \stackrel{ST}{2x}~~ = ~~\stackrel{RS}{4x-4} \\\\\\ -2x=-4\implies x=\cfrac{-4}{-2}\implies \boxed{x=2}~\hfill \stackrel{4(2)~~ - ~~4}{RS=8}[/tex]