To find dy/dx for y = 1/3x³ + 7x, we need to take the derivative with respect to x using the power rule and the sum rule, which gives dy/dx = x² + 7.
The given equation is y = 1/3x³ + 7x. To find dy/dx, we need to take the derivative of y with respect to x. We can do this by using the power rule and the sum rule of differentiation.
Using the power rule, the derivative of x³ is 3x². Using the sum rule, the derivative of 7x is 7. Therefore, the derivative of y with respect to x is:
dy/dx = d/dx (1/3x³ + 7x)
= d/dx (1/3x³) + d/dx (7x)
= (1/3) d/dx (x³) + 7 d/dx (x)
= (1/3) (3x²) + 7
= x² + 7
Hence, we have found that the derivative of y with respect to x is x² + 7.
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A park is to be designed as a circle. A straight walkway will intersect the fence of the
park twice, requiring gates at each location. The city planner draws the circular park
and the walkway on a coordinate plane, with the equation
x² + y² - 4x = 9 for the circular park and the equation y = 2x modeling the
walkway. Write an ordered pair that represents the location of the gates in the third
quadrant.
o find the coordinates of the gates in the third quadrant, we need to find the points where the circle and the line intersect in the third quadrant.
Substituting y = 2x into x² + y² - 4x = 9, we get:
x² + (2x)² - 4x = 9
5x² - 4x - 9 = 0
Using the quadratic formula, we find:
x = (-(-4) ± √((-4)² - 4(5)(-9))) / (2(5))
x = (4 ± √136) / 10
We can discard the positive root since it is in the first quadrant. The negative root corresponds to the x-coordinate of the point of intersection in the third quadrant:
x = (4 - √136) / 10 ≈ -0.433
Substituting this value into y = 2x, we get:
y = 2(-0.433) ≈ -0.866
Therefore, the ordered pair that represents the location of the gates in the third quadrant is (-0.433, -0.866).
What is question asking??? All I need is one example and I get the rest I just don’t understand the assignment
So basically you have to drag an angle to the box, example drag DFA to the box, then add what angle it is equal to. You have to calculate the angle and drag it opposite to the DFA. For DFA, it will be 58.
suppose x is a random variable with mean mu and standard deviation sigma. If a large number of trials are observed, at least what percentage of these values is expected to lie between mu minus 2 sigma and mu plus 2 sigma?
At least 95% of the observed values are expected to lie between mu minus 2 sigma and mu plus 2 sigma.
This is because of the empirical rule, also known as the 68-95-99.7 rule, which states that in a normal distribution, approximately 68% of the observations will fall within one standard deviation of the mean, about 95% of the observations will fall within two standard deviations of the mean, and around 99.7% of the observations will fall within three standard deviations of the mean.
In this case, we are given that x has mean mu and standard deviation sigma. Therefore, about 95% of the values of x are expected to lie between mu minus 2 sigma and mu plus 2 sigma, as this interval covers two standard deviations on either side of the mean.
Mathematically, we can express this as:
P(mu - 2sigma < x < mu + 2sigma) ≈ 0.95
where P is the probability that x falls within the interval mu - 2sigma to mu + 2sigma.
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What is the surface area of the square pyramid with base edge of 10 millimeters and a face height of 18 millimeters?
180 mm2
600 mm2
460 mm2
360 mm2
The surface area of the square base pyramid is calculated to be equal to 460 square millimetres.
How to calculate for the total surface area of the square base pyramidarea of one triangle face = 1/2 × 10 mm × 18 = 90 mm² mm
area of the four triangle faces = 4 × 90 mm² = 360 mm²
area of the square base = 10 mm × 10 mm = 100 mm²
surface area of the square base pyramid = 360 mm² + 100 mm²
surface area of the square base pyramid = 460 mm²
Therefore, the surface area of the square base pyramid is calculated to be equal to 460 square millimetres.
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If 3 quarts is greater then 4 prints is that an equivalent measure
If 3 quarts is greater than 4prints, then the measure is not equivalent.
What is equivalent measurement?Equivalent units can be used to convert different units to the same unit for comparison. Equivalent means equal. For example , 1 kilogram is equal to 1,000 grams.
For example,
3 teaspoons = 1 tablespoon.
4 tablespoons = 1/4 cup.
5 tablespoons + 1 teaspoon = 1/3 cup.
8 tablespoons = 1/2 cup.
1 quart = 2pints
therefore 3 quarts = 2×3 = 6pints
therefore the statement that 3 quarter is greater than 4 prints is true and not an equivalent measure.
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Aadya has 143 stamps; she gives away 11 stamps and divides the remaining equally into groups.
Sumit has 220 stamps; he gives away 11 stamps and divides the remaining equally into groups.
They end up with the same number of groups.
(a) What is the number of groups?
(b)what is the No. of stamps in each of their groups
Answer:
a) The number of groups are 11.
b) For Aadya, there are 12 stamps in each group. For Sumit, there are 19 stamps per group.
Step-by-step explanation:
Aadya: 143 - 11 = 132 stamps.
Sumit: 220 - 11 = 209 stamps.
Greatest Common Factor of 132 and 209 = 11 group for both
Aadya: Let a = # of stamps in each group.; 11a = 132; a = 12 stamps per group
Sumit: Let s = # of stamps in each group.; 11s = 209; s = 19 stamps per group.
Julia just lit a new candle and then let it burn all the way down to nothing. The candle burned at a rate of 0.75 inches per hour and its initial length was 9 inches. Write an equation for
L
,
L, in terms of
t
,
t, representing the length of the candle remaining unburned, in inches,
t
t hours after the candle was lit.
The required equation for the length of the candle is L = 9-0.75t.
Given that a 9-inch candle burns at the rate of 0.75 in per hour we need to determine the equation for the length L of the candle when it burns for t hours,
So, if the candle is burning at the rate of 0.75 in per hour so after burning for t hour the length decreases by 0.75t, from the original length i.e 9 in
So, we can write the equation as =
L = 9-0.75t
Hence the required equation for the length of the candle is L = 9-0.75t.
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Please help I am giving a lot of points
A circle has been dissected into 16 congruent sectors. The base of one sector is 1. 56 units, and its height is 3. 92 units. Using the area of a triangle formula, what is the approximate area of the circle?
circle A is dissected into 16 congruent sectors, one sector is highlighted
27. 52 units2
48. 25 units2
48. 92 units2
76. 44 units2
The closest answer choice is [tex]27.52 units^2.[/tex]
The area of the circle, we need to find the area of one sector and then multiply it by 16 since there are 16 congruent sectors.
To find the area of one sector, we use the formula:
[tex]Area of sector = (angle/360) * \pi*r^2[/tex]
Since we know the base and height of the highlighted sector, we can use the Pythagorean theorem to find the radius of the circle:
[tex]r^2 = (1.56/2)^2 + (3.92)^2[/tex]
r ≈ 3.969 units
Now we can find the angle of one sector using the formula:
angle = (base/radius) x 180/π
angle ≈ 22.5 degrees
Plugging in the values for angle and radius in the area of sector formula, we get:
[tex]Area of sector =(22.5/360) *\pi (3.969)^2[/tex]
Area of sector ≈ 0.491π
Multiplying this by 16, we get the approximate area of the circle:
Approximate area of circle ≈ 16 x 0.491π
Approximate area of circle ≈ 7.8π
Using a calculator to approximate π as 3.14, we get:
Approximate area of circle ≈ [tex]24.46 units^2[/tex]
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1. bona drives tor 3 hours at 44mph. clare drives 144 mies in 4 hours. how
for would bena travel if she drove for 3 hours at the same speed os
claire
2. janet and andrew leave their home at the same time. janet has 60
milles to travel and drives at 40 mph. andrew have 80 miles to travel
and also drives at 40 mph
a) how long does janets journey take?
(b) how much longer does andrew spend driving than janeta
1) Bona can drive 108 miles if she drove for 3 hour at the same speed as Claire.
2) Janet would take 1.5 hour to complete the journey and Andrew spend half hour more driving than Janet.
1) Bona speed is 44mph
Time taken by Bona is 3 hour
distance travelled by Claire is 144 miles
Time taken by Claire is 4 hour
Claire's speed = distance / time
Claire's speed = 144/4
Claire's speed = 36 mph
Distance travelled by Bona = speed × time
Distance travelled by Bona = 36 × 3
Distance travelled by Bona = 108 miles
2) Janet distance = 60 miles
Janet speed = 40 mph
Time taken by Janet = distance / speed
Time taken by Janet = 60/40
Time taken by Janet = 1.5 hour
Janet would take 1.5 hour to complete the journey
Andrew distance = 80 miles
Andrew speed = 40 mph
Time taken by Andrew = distance / speed
Time taken by Andrew = 80/40
Time taken by Andrew= 2 hour
Andrew spend half hour more driving than Janet.
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Find the value of x. round to the nearest degree.
14
5
x =
degrees
anybody knows the answer to this ?
x is approximately 20.5 degrees when rounded to the nearest degree. Therefore, x ≈ 21 degrees.
General process of solving for an unknown angle.
1. Determine the type of angle: Determine whether the angle is a right angle (90 degrees), acute (less than 90 degrees), or obtuse (greater than 90 degrees).
2. Use geometric properties: If there are geometric properties or relationships given in the problem, such as angles formed by parallel lines or within a triangle, apply those properties to find the value of x.
3. Apply trigonometric functions: If the problem involves trigonometry, use sine, cosine, or tangent functions along with the given information to solve for x.
4. Apply algebraic equations: If there is an algebraic equation involving x, set up the equation and solve for x by isolating it on one side of the equation.
To find the value of x in the given triangle, we can use the inverse tangent function, which is tan^-1.
tan(x) = opposite/adjacent
tan(x) = 5/14
To isolate x, we take the inverse tangent of both sides:
x = tan^-1(5/14)
Using a calculator, we can find that x is approximately 20.5 degrees when rounded to the nearest degree. Therefore, x ≈ 21 degrees.
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In a recent election 59% of people supported re-electing the incumbent. Suppose a poll is done of 1230 people. If we used the normal as an approximation to the binomial, what would the mean and standard deviation be? Please show formulas used in excel
The mean is 725.7 and the standard deviation is 13.55.
To find the mean and standard deviation using the normal approximation to the binomial, we will use the following formulas in Excel:
Mean = np
Standard Deviation = sqrt(np(1-p))
Where n = sample size, p = proportion of success, and sqrt = square root.
Using the information given in the question, we can plug in the values:
n = 1230
p = 0.59
Mean = np = 1230*0.59 = 725.7
Standard Deviation = sqrt(np(1-p)) = sqrt(1230*0.59*(1-0.59)) = 13.55
Therefore, the mean is 725.7 and the standard deviation is 13.55.
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Julia works at a music store. One of her jobs is to stock new CDs on the shelf. A recent order arrived with 215 classical CDs, 125 jazz CDs, and 330 soft rock CDs. How many groups will Julia use to arrange all of the CDs?
Julia will use 10 groups to arrange all of the CDs.
To determine the number of groups Julia will use to arrange all of the CDs, we need to find the greatest common divisor of the numbers 215, 125, and 330.
First, we can check if any of the numbers are divisible by 5:
215 is not divisible by 5
125 is divisible by 5 (125 ÷ 5 = 25)
330 is divisible by 5 (330 ÷ 5 = 66)
Now we divide 125 and 330 by 5:
125 ÷ 5 = 25
330 ÷ 5 = 66
Next, we check if any of the numbers are divisible by 2:
25 is not divisible by 2
66 is divisible by 2 (66 ÷ 2 = 33)
Now we divide 66 by 2:
66 ÷ 2 = 33
Therefore, the greatest common divisor of 215, 125, and 330 is 5 × 2 = 10.
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State the null and alternative hypotheses you would use to test the following situation. The average time it takes for a person to experience pain relief from a certain pain reliever is 15 minutes. A new ingredient is added to help speed up pain relief and an experiment is conducted to test whether the new product does indeed speed up pain relief. What are the appropriate null and alternative hypotheses for the experiment
The appropriate null and alternative hypotheses for the experiment are:
Null hypothesis (H0): According to null hypothesis, the average time it takes for a person to experience pain relief from the new pain reliever is not beyond 15 minutes.
Alternative hypothesis (Ha): According to Alternative hypothesis, the average time it takes for a person to experience pain relief from the new pain reliever is significantly less than 15 minutes, indicating that the new ingredient does speed up pain relief.
The alternative and null hypotheses can be written as follows in symbols:
H0: = 15 (where μ is the population mean time for pain relief from the new pain reliever)
Ha: μ < 15
The one-tailed hypothesis test assumes that the new component of the painkiller can only reduce the duration of pain alleviation, not lengthen it. As a result, the rejection region will be in the left tail of the distribution, while the alternative hypothesis is one-tailed to the left.
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3.
What does limit as x goes to infinity of the quotient of f of x and g of x equals 5 show? (4 points)
g(x) grows faster than f(x) as x goes to infinity.
f(x) and g(x) grow at the same rate as x goes to infinity.
f(x) grows faster than g(x) as x goes to infinity.
L'Hôpital's Rule must be used to determine the true limit value
The statement "limit as x goes to infinity of the quotient of f of x and g of x equals 5" means that as x gets larger and larger, the ratio of f(x) to g(x) approaches 5.
If f(x) grows faster than g(x) as x goes to infinity, then the ratio of f(x) to g(x) would approach infinity, not 5.
If f(x) and g(x) grow at the same rate as x goes to infinity, then the ratio of f(x) to g(x) would approach a constant value, not necessarily 5.
Therefore, the statement "limit as x goes to infinity of the quotient of f of x and g of x equals 5" shows that g(x) grows faster than f(x) as x goes to infinity. L'Hôpital's Rule is not necessary to determine this conclusion.
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Use the nth-term test for divergence to show that the series is divergent, or state that the test is inconclusive.
∑ 1/n+13
Using the nth-term test for divergence on the series ∑ 1/n+13 is inconclusive. However, by comparing the series to the divergent harmonic series, we can conclude that ∑ 1/n+13 is also divergent.
We can use the nth-term test for divergence to determine the convergence or divergence of the series
lim n → ∞ (1/n+13) = 0
Since the limit of the nth term is 0, the nth-term test is inconclusive, and we cannot determine the convergence or divergence of the series using this test.
However, we can use the comparison test to show that the series diverges. We can compare the given series to the harmonic series, which we know diverges
1/1 + 1/2 + 1/3 + ...
Since each term of the given series is less than the corresponding term of the harmonic series, the given series must also diverge. Therefore, the series ∑ 1/n+13 is divergent.
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In the derivation of the quadratic formula by completing the square, the equation mc032-1. Jpgis created by forming a perfect square trinomial. What is the result of applying the square root property of equality to this equation?.
The result of applying the square root property of equality to this equation is x = (-b ± √(b² - 4ac)) / (2a)
If we apply the square root property of equality to the equation (x + (b/2a))² = (-4ac + b²)/(4a²), we get:
x + (b/2a) = ±√[(-4ac + b²)/(4a²)]
Next, we can simplify the expression under the square root:
√[(-4ac + b²)/(4a²)] = √(-4ac + b²)/2a
Now, we can substitute this expression back into our original equation:
x + (b/2a) = ±√(-4ac + b²)/2a
Finally, we can isolate x by subtracting (b/2a) from both sides:
x = (-b ± √(b² - 4ac)) / (2a)
This is the quadratic formula, which gives us the solutions for the quadratic equation ax² + bx + c = 0. By completing the square, we have derived this formula from the original quadratic equation.
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Complete question is:
In the derivation of the quadratic formula by completing the square, the equation (x+ (b/2a))² =(-4ac+b²)/(4a²) is created by forming a perfect square trinomial What is the result of applying the square root property of equality to this equation?
Ben earns $12. 50 per hour and $6 for each delivery he makes. He wants to earn $168 in an 8-hour workday.
Part A) Which equation could you use to solve for the least number of deliveries he must make to reach his goal?
Part B) What is the least number of deliveries he must make to reach his goal?
Answer:
ben earns $9 per hour and $6 for each delivery he makes
Step-by-step explanation:
if an i statement is true, what is the truth value of its corresponding a statement? true false logically undetermined
If an 'I' statement is true, then the truth value of its corresponding E statement is equals to the true. So, option(a) is right answer of this problem.
Truth value : The truth value of a statement is either true or false, depending on whether the logic makes sense or not. Let us assume two statements, p : Ram goes to school daily
q : Ram will get good marks
We use implication for determining value. In logic, implication is relationship between different propositions in which the second proposition is a logical consequence of the first.
If p is true and q is true, then (p implies q) is true.If p is true and q is false, then (p implies) must be false.first case : If Ram does not go to school daily then he will get good marks. That is p is false and q is true, ¬ p --> q ⇒ true ( using implication rule)
second case : If Ram does not go to school daily then he will not get good marks. That is p is false and q is false , ¬ p --> ¬ q ⇒ true ( using implication rule)
Hence, required value of statement is true.
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Suppose a mouse is placed in the maze at the right. if each desicion about direction is made at random, create a simulation to determine the probability that the mouse will find its way out before coming to a dead end or going out in the opening.
The probability of the mouse finding its way out before reaching a dead end or going out in the opening can be estimated by dividing the number of successful outcomes by the total number of trials in the sample.
To create a simulation to determine the probability that the mouse will find its way out before coming to a dead end or going out in the opening, we can follow these steps:
Create a model of the maze in a programming language such as Python.Define the starting position of the mouse as the position on the right side of the maze.Define the exit position of the maze as the position on the left side of the maze.Randomly choose a direction for the mouse to move in (up, down, left or right).Check if the chosen direction leads to a dead end or out of the maze. If it does, return a failure outcome.If the chosen direction leads to a viable path, move the mouse to that position and repeat steps 4-6 until the mouse either reaches the exit or gets stuck in a dead end.Repeat steps 2-6 multiple times to generate a sufficient sample size.Calculate the proportion of successful outcomes (i.e. the mouse finding its way out before reaching a dead end or going out in the opening) from the generated sample.The probability of the mouse finding its way out before reaching a dead end or going out in the opening can be estimated by dividing the number of successful outcomes by the total number of trials in the sample. This simulation approach can help us understand the probability of success in a random maze environment, and also explore the impact of various factors such as maze complexity, size and starting position of the mouse on the outcome.
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if I draw a marble 48 times a white marble is selected 35 times ana a yellow one is selected 13 times what is the probability of the next one to be yellow
A 13%
B 27%
C 51%
D 63%
If we are drawing without replacement, the probability is approximately 27.7%, which is closest to option B: 27%.
What is the probability that the next marble is yellow?The probability of drawing a yellow marble on the next draw depends on whether we are drawing with or without replacement.
If we are drawing without replacement, then the probability of drawing a yellow marble is 13 out of the remaining 47 marbles, since we have already drawn 35 white marbles and 13 yellow marbles out of the 48 total marbles.
If we are drawing with replacement, then the probability of drawing a yellow marble on the next draw is still 1/3, or approximately 33.3%.
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A rectangular playing field lies in the interior of an elliptical track that is 50 yards wide and 110 yards long. What is the width of of the rectangular playing field if the width is located 15 yards from either vertex?
The width of the rectangular playing field is approximately 50 yards if the width is located 15 yards from either vertex.
To solve the problem, we can draw a diagram and use the properties of ellipses.
First, we note that the major axis of the ellipse is 110 yards and the minor axis is 50 yards. We can find the distance between the two foci of the ellipse using the formula c^2 = a^2 - b^2, where c is the distance between the foci, and a and b are the lengths of the semi-major and semi-minor axes.
c^2 = 110^2 - 50^2
c^2 = 10800
c ≈ 104.0
Next, we draw the two foci of the ellipse and the rectangle as shown in the diagram below. We are given that the width of the rectangle is 30 yards (15 yards from either vertex). x be the length of the rectangle.
A B
+-------+-------+
/ \
/ \
/ \
C D
\ /
\ /
\ /
+-------+-------+
E F
We can see that the length of the rectangle is equal to the distance between points A and B, and the width of the rectangle is equal to the distance between points C and D. Using the Pythagorean theorem, we can find the length of the rectangle.
AB^2 = AE^2 + EB^2
AB^2 = (a/2)^2 + (c - b/2)^2
AB^2 = (55)^2 + (104 - 15)^2
AB^2 = 3025 + 7225
AB = sqrt(10250)
AB ≈ 101.2
Therefore, the length of the rectangle is approximately 101.2 yards.
To find the width of the rectangle, we can use the fact that the distance between points C and D is equal to twice the distance between the center of the ellipse and the minor axis. The center of the ellipse is the midpoint of the major axis, and the distance from the center to the minor axis is 25 yards.
CD = 2 * 25 = 50
Therefore, the width of the rectangle is approximately 50 yards.
In summary, the width of the rectangular playing field is approximately 50 yards if the width is located 15 yards from either vertex.
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(2) Use the Comparison Test or the Limit Comparison Test to determine the convergence or divergence of the following series. Justify your answer. 23-1 5k - 1 k=1 (3) Use the Integral Test to determine the convergence or divergence of the following series. Justify your answer. Ink k k=1
2)By using Comparison Test,the series 23-1 5k-1 k=1,is divergent.
3)By using Integral Test the series Ink k k=1 is divergent.
2) To determine the convergence or divergence of the series 23-1 5k - 1 k=1:
For the first series, 23-1 5k-1 k=1, we can use the Limit Comparison Test.
Let's compare it to the series 5k-1 k=1.
We take the limit as k approaches infinity of the ratio of the two series:
lim(k->∞) [(23-1 5k-1) / (5k-1)] = lim(k->∞) [23 / 5] = 23/5
Since this limit is finite and positive, and the series 5k-1 diverges (as it is a p-series with p=1),
we can conclude that the given series also diverges.
3)To determine the convergence or divergence of the series Ink k k=1:
For the second series, Ink k=1, we can use the Integral Test.
We need to check if the following improper integral converges or diverges:
∫(1 to ∞) ln(x) dx
Integrating by parts, we get:
∫(1 to ∞) ln(x) dx = [xln(x) - x]1∞ + ∫(1 to ∞) dx/x
The first term evaluates to -∞, and the second term is the divergent harmonic series.
Therefore, the improper integral and the series both diverge.
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Please Help Quick!!!!
The tip of a pendulum swings through an arc whose length is 8 centimeters. If the radius of the pendulum is 26 centimeters, then what radian angle did the tip rotate through? Round your answer to the nearest hundredth
Answer: Therefore, the tip of the pendulum rotated through an angle of approximately 0.31 radians.
Step-by-step explanation:
The length of the arc traveled by the tip of the pendulum is 8 centimeters, and the radius of the pendulum is 26 centimeters. We can use the formula for the length of an arc of a circle to find the radian angle rotated through:
Length of arc = radius * angle in radians
Solving for the angle in radians, we get:
Angle in radians = Length of arc / radius
Plugging in the given values, we get:
Angle in radians = 8 / 26
Simplifying this expression, we get:
Angle in radians = 0.30769...
Rounding this value to the nearest hundredth, we get:
Angle in radians = 0.31
Therefore, the tip of the pendulum rotated through an angle of approximately 0.31 radians.
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The temperature at sunrise was T degrees. By noon the temperature had tripled. By sunset, the temperature was only half of what the
temperature was at noon.
Which expression shows the temperature at sunset in terms of T?
OA (T+3) = Ź
(T+3)
2
Ос. 37 = 5
1 / 2
3. 37 를
D
The expression that shows the temperature at sunset in terms of T is 3T/2.
Let's call the temperature at sunrise T. According to the problem statement, the temperature tripled from sunrise to noon, so the temperature at noon is 3T.
Then, from noon to sunset, the temperature halved, so the temperature at sunset is (1/2) of the temperature at noon, or (1/2)(3T), which simplifies to 3T/2. Therefore, the expression that shows the temperature at sunset in terms of T is 3T/2.
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A baseballâ player's batting average is 0. 343â, which can be interpreted as the probability that he got a hit each time at bat. â Thus, the probability that he did not get a hit is 1â0. 343=0. 657. Assume that the occurrence of a hit in any givenâ at-bat has no effect on the probability of a hit in otherâ at-bats. In oneâ game, the player had 5 âat-bats. What is the probability that he had 3 âhits? What expression can be used to calculate theâ probability?
0.135 or 13.5% is the probability that he had 3 âhits
The probability that the player had 3 hits in 5 at-bats can be calculated using the binomial probability formula, which is:
P(x) = (nCx) * p^x * (1-p)^(n-x)
where:
- P(x) is the probability of getting x hits
- n is the number of at-bats (in this case, 5)
- x is the number of hits we want to find the probability for (in this case, 3)
- p is the probability of getting a hit in one at-bat (in this case, 0.343)
- (1-p) is the probability of not getting a hit in one at-bat (in this case, 0.657)
Plugging in the values, we get:
P(3) = (5C3) * 0.343^3 * 0.657^(5-3)
P(3) = (10) * 0.039304527 * 0.4305961
P(3) = 0.134912947
Therefore, the probability that the player had 3 hits in 5 at-bats is approximately 0.135 or 13.5%.
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In 1990, Jane became a real estate agent. Eight years later, she sold a house for $144,000. Eleven years later, she sold the same house for $245,000. Write an equation that represents the value of the house (V) related to the number of years (t) since Jane became a real estate agent. (Hint: Be careful! The second sale of the house was 11 years after the first sale which was 8 years after she became a real estate agent! That means the second sale took place 19 years after she became an agent!)
Let's break down the information given in the problem:
- Jane became a real estate agent in 1990.
- She sold a house 8 years later (in 1998) for $144,000.
- She sold the same house 11 years after that sale (in 2009), which is 19 years after she became an agent, for $245,000.
To write an equation that represents the value of the house (V) related to the number of years (t) since Jane became a real estate agent, we can use the information from the two sales to find the rate of change in the value of the house over time. We can use this rate of change to write an equation in point-slope form:
V - V1 = m(t - t1)
where V1 is the value of the house at time t1, m is the rate of change in the value of the house, and t is the time since Jane became a real estate agent.
Using the two sales, we can find the rate of change in the value of the house as follows:
m = (V2 - V1) / (t2 - t1)
where V2 is the value of the house at the second sale, t2 is the time of the second sale (19 years after Jane became an agent), V1 is the value of the house at the first sale, and t1 is the time of the first sale (8 years after Jane became an agent).
Substituting the given values, we get:
m = ($245,000 - $144,000) / (19 - 8) = $10,100 per year
Now we can use the point-slope form equation to find the value of the house at any time t since Jane became a real estate agent. Let's choose 1990 as our initial time (t1), so V1 = $0:
V - 0 = $10,100 (t - 0)
Simplifying, we get:
V = $10,100t
Therefore, the equation that represents the value of the house (V) related to the number of years (t) since Jane became a real estate agent is V = $10,100t. Note that this equation assumes a constant rate of change in the value of the house over time, which may not be accurate in real life.
Exl) Solve exercise 3 on the first page of the handout; that is, find y" if 2.23 – 3y = 8. Then, answer the following question. What equation did you obtain after differentiating both sides of the given equation with respect to x? ENTER dy/dx or y'where needed, enter a power using the symbol^, for example enter was x^3, NO SPACES: fill in blank
To solve exercise 3 on the first page of the handout, we need to first isolate y in the given equation 2.23 - 3y = 8, which gives us y = -1.59.
To find y", we need to differentiate both sides of the equation with respect to x twice. The first derivative gives us:
-3(dy/dx) = 0
Simplifying, we get dy/dx = 0.
Differentiating again, we get:
-3d^2y/dx^2) = 0
Simplifying, we get d^2y/dx^2 = 0.
Therefore, the equation we obtain after differentiating both sides of the given equation with respect to x is d^2y/dx^2 = 0, which is the second derivative of y with respect to x.
To solve the equation given on the first page of the handout, 2.23 - 3y = 8, first isolate y:
1. Subtract 2.23 from both sides: -3y = 5.77
2. Divide both sides by -3: y = -5.77/3
Your answer: y = -5.77/3
The original equation doesn't have any x terms, so differentiation with respect to x is not applicable in this case.
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How to show (sin 7x)/sin x = 64(cos x)^6 - 80(cos x)^4 +24(cos x)^2 - 1 ?
Following shows (sin 7x)/sin x = 64(cos x)^6 - 80(cos x)^4 +24(cos x)^2 - 1:
12sin^3 x - 8sin^
To show that (sin 7x)/sin x is equal to 64(cos x)^6 - 80(cos x)^4 + 24(cos x)^2 - 1, we can use trigonometric identities and algebraic manipulation.
Let's start with the left-hand side of the equation:
(sin 7x)/sin x
Using the trigonometric identity for sin(A + B):
sin(A + B) = sin A cos B + cos A sin B
We can rewrite sin 7x as sin (6x + x):
sin (6x + x) = sin 6x cos x + cos 6x sin x
Now we can substitute sin 7x with sin 6x cos x + cos 6x sin x:
(sin 6x cos x + cos 6x sin x)/sin x
Next, we can simplify this expression by dividing both terms by sin x:
(sin 6x cos x)/sin x + (cos 6x sin x)/sin x
The sin x term cancels out, leaving us with:
sin 6x cos x + cos 6x
Now, we can use the double-angle identity for sin 2A:
sin 2A = 2sin A cos A
To rewrite sin 6x cos x, we can treat it as sin 2A with A = 3x:
sin 6x cos x = 2sin 3x cos 3x
Next, we can use the triple-angle identity for sin 3A:
sin 3A = 3sin A - 4sin^3 A
To rewrite sin 3x, we can treat it as sin A with A = x:
sin 3x = 3sin x - 4sin^3 x
Substituting this into our expression:
2sin 3x cos 3x = 2(3sin x - 4sin^3 x) cos 3x
Expanding further:
= 6sin x cos 3x - 8sin^3 x cos 3x
Now, we can use the double-angle identity for cos 2A:
cos 2A = cos^2 A - sin^2 A
To rewrite cos 3x, we can treat it as cos A with A = x:
cos 3x = cos^2 x - sin^2 x
Substituting this into our expression:
6sin x cos 3x - 8sin^3 x cos 3x = 6sin x (cos^2 x - sin^2 x) - 8sin^3 x (cos^2 x - sin^2 x)
Expanding further:
= 6sin x cos^2 x - 6sin x sin^2 x - 8sin^3 x cos^2 x + 8sin^3 x sin^2 x
Now, we can use the Pythagorean identity for sin^2 x + cos^2 x:
sin^2 x + cos^2 x = 1
To rewrite sin^2 x, we can subtract cos^2 x from both sides:
sin^2 x = 1 - cos^2 x
Substituting this back into our expression:
= 6sin x (cos^2 x - (1 - sin^2 x)) - 8sin^3 x cos^2 x + 8sin^3 x sin^2 x
= 6sin x (2sin^2 x) - 8sin^3 x cos^2 x + 8sin^3 x sin^2 x
= 12sin^3 x - 8sin^
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Identify the measure of arc FE⏜ given the measure of arc FGC⏜ is 220∘
The value of the angle of the arc FE⏜ is calculated as: 20°
How to find the angle at the arc?The angle of an arc is identified by its two endpoints. The measure of an arc angle is found by dividing the arc length by the circle's circumference, then multiplying by 360 degrees. Formulas for calculating arcs and angles vary based on where they are in reference to the circle.
Now, from the given image, we see that:
FGC⏜ = 220°
∠B = 30°
∠OEC = ∠OCE = 30°
CDE⏜ = 220°
Thus:
FE⏜ = 360° - 220° - 120°
FE⏜ = 20°
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