The parametrization of the tangent line to the function f(x) = ln(x) - 7/15x^3 at the point (1,-46/15) is r(t) = <1, -46/15> + t<1, -2/3>.
To find the tangent line at a point, we need the slope of the tangent line, which is the derivative of the function evaluated at that point. So, we first find the derivative of f(x):
f'(x) = 1/x - 7/5 x^2
Then, we evaluate f'(1) to find the slope at x = 1:
f'(1) = 1/1 - 7/5(1)^2 = -2/5
Thus, the slope of the tangent line is -2/5. We also know that the point of tangency is (1,-46/15), so we can use the point-slope form to find the equation of the tangent line:
y - (-46/15) = (-2/5)(x - 1)
Simplifying, we get:
y = (-2/5)x - 16/3
Now we can write the parametrization of the tangent line as r(t) = <1, -46/15> + t<1, -2/3>. This is because the direction vector of the tangent line is <1, -2/3>, which is the same as the slope of the line, and the point on the line is (1,-46/15).
So, to get the equation of the line in vector form, we start with the point <1, -46/15>, and add a scalar multiple of the direction vector <1, -2/3>. Thus, the parametrization of the tangent line is r(t) = <1, -46/15> + t<1, -2/3>.
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The parametrization of the tangent line to the function f(x) = ln(x) - 7/15x^3 at the point (1,-46/15) is r(t) = <1, -46/15> + t<1, -2/3>.
To find the tangent line at a point, we need the slope of the tangent line, which is the derivative of the function evaluated at that point. So, we first find the derivative of f(x):
f'(x) = 1/x - 7/5 x^2
Then, we evaluate f'(1) to find the slope at x = 1:
f'(1) = 1/1 - 7/5(1)^2 = -2/5
Thus, the slope of the tangent line is -2/5. We also know that the point of tangency is (1,-46/15), so we can use the point-slope form to find the equation of the tangent line:
y - (-46/15) = (-2/5)(x - 1)
Simplifying, we get:
y = (-2/5)x - 16/3
Now we can write the parametrization of the tangent line as r(t) = <1, -46/15> + t<1, -2/3>. This is because the direction vector of the tangent line is <1, -2/3>, which is the same as the slope of the line, and the point on the line is (1,-46/15).
So, to get the equation of the line in vector form, we start with the point <1, -46/15>, and add a scalar multiple of the direction vector <1, -2/3>. Thus, the parametrization of the tangent line is r(t) = <1, -46/15> + t<1, -2/3>.
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find an equation of the plane through the three points given p=(5,0,0) q=(6,-2,4)
The equation of the plane that passes through the three given points P(5,0,0) and Q(6,-2,4) is 2y + z + D/8 = 0, where D is a constant that depends on the specific plane.
The equation of the plane that passes through the three given points P(5,0,0) and Q(6,-2,4) can be represented as Ax + By + Cz + D = 0, where A, B, C, and D are constants that need to be determined.
Step 1: Find two vectors on the plane
We can find two vectors on the plane by subtracting the coordinates of one point from the other. Let's take vector PQ as the first vector, which is the difference between the coordinates of points P and Q.
PQ = Q - P = (6, -2, 4) - (5, 0, 0) = (1, -2, 4)
Step 2: Find the normal vector of the plane
The normal vector of the plane is perpendicular to the plane and can be found by taking the cross product of the two vectors obtained in Step 1.
Normal vector = PQ x PR, where PR is any other vector on the plane
We can choose vector PR as (1, 0, 0) for convenience.
PR = R - P = (1, 0, 0) - (5, 0, 0) = (-4, 0, 0)
Taking the cross product of PQ and PR:
PQ x PR = (1, -2, 4) x (-4, 0, 0) = (0, 16, 8)
So, the normal vector of the plane is (0, 16, 8).
Step 3: Write the equation of the plane
Using the normal vector and one of the given points (P), we can now write the equation of the plane.
The equation of the plane is given by:
Ax + By + Cz + D = 0
Substituting the values of the normal vector and the coordinates of point P into the equation, we get:
0x + 16y + 8z + D = 0
We can further simplify this equation by dividing by 8:
2y + z + D/8 = 0
Therefore, the equation of the plane that passes through the three given points P(5,0,0) and Q(6,-2,4) is 2y + z + D/8 = 0, where D is a constant that depends on the specific plane.
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Multiply. (w-2v)(w+2v) simply your answer
Answer:
w^2-4v^2
Step-by-step explanation:
w*w=w^2
w*2v=2vw
-2v*w=-2vw
-2v*2v=-4v^2
w^2+2vw-2vw-4v^2
w^2-4v^2
A Stock Clerk's income is $832.00 a month and his total expenses are $668. How much money does he have left for savings?
Answer:
$164
Step-by-step explanation:
you take the clerk's income ($832.00) and subtract it with the total expenses ($668) to get the savings
The sum of three numbers $x$, $y$, $z$ is $165$. When the smallest number $x$ is multiplied by $7$, the result is $n$. The value $n$ is obtained by subtracting $9$ from the largest number $y$. This number $n$ also results by adding $9$ to the third number $z$. What is the product of the three numbers?
Hint: It's not 12295
The product of the three numbers under the given circumstances is 49,483.
How are products of numbers determined?Let's start by setting up the equations based on the given information:
x + y + z = 165 (equation 1)
7x = n (equation 2)
y - 9 = n (equation 3)
z + 9 = n (equation 4)
We want to find the product of x, y, and z, which is simply:
x * y * z
We can use equations 2, 3, and 4 to substitute n in terms of y and z:
7x = y - 9 (substituting equation 3)
7x = z + 9 (substituting equation 4)
Now we can substitute these expressions for y and z into equation 1 to get an equation in terms of x:
x + (7x + 9) + (7x - 9) = 165
15x = 165
x = 11
Substituting x = 11 into equations 2, 3, and 4, we get:
7(11) = n
n = 68
y = n + 9 = 68 + 9 = 77
z = n - 9 = 68 - 9 = 59
Now we can calculate the product of x, y, and z:
x * y * z = 11 * 77 * 59 = 49,483
Therefore, the product of the three numbers is 49,483.
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the volume of a cube decreases at a rate of 0.4 ft^3/min. what is the rate of change on the side length when the side lengths are 12 feet?
Rate of change in the side length when the side lengths are 12 feet is approximately -0.0009259 ft/min.
Explanation:-
To find the rate of change in the side length of the cube when the side lengths are 12 feet and the volume decreases at a rate of 0.4 ft³/min, follow these steps:
Step 1: The formula for the volume of a cube.
Volume (V) = side length³, or V = s³
Step 2: Differentiate both sides with respect to time (t) to find the relationship between the rates of change.
dV/dt = 3s²(ds/dt)
Step 3: Plug in the given information: dV/dt = -0.4 ft³/min (since the volume is decreasing), and s = 12 feet.
-0.4 = 3(12²)(ds/dt)
Step 4: Solve for ds/dt, the rate of change in the side length.
-0.4 = 3(144)(ds/dt)
-0.4 = 432(ds/dt)
ds/dt = -0.4/432
Step 5: Simplify the expression.
ds/dt ≈ -0.0009259 ft/min
So, the rate of change in the side length when the side lengths are 12 feet is approximately -0.0009259 ft/min.
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Start at (-2, 4) and move 3 units to the right and 5 units down. What is the new location?
5/6 + 2/3 = ?
your answer
Answer:
3/2 or 1.5 or 1 1/2
Step-by-step explanation:
5/6 + 2/3 = ?
5/6 + 4/6 =
9/6
semplify
3/2 or 1.5 or 1 1/2
The degrees of freedom for the sample variance A.are equal to the sample size B.are equal to the sample size C.can vary between - [infinity] and + [infinity] D.both B and C
The degrees of freedom for the sample variance can vary between - [infinity] and + [infinity]. This means that the number of degrees of freedom is not dependent on the sample size, but rather on the amount of variance in the data.
The degrees of freedom for sample variance A. is equal to the sample size minus 1. This means that the correct answer is not provided in your given options. To clarify, let's define the terms:
1. Degrees of freedom: The number of independent values in a statistical calculation that are free to vary.
2. Variance: A measure of dispersion that represents the average squared difference between the values in a dataset and the mean of the dataset.
3. Sample size: The number of observations in a sample.
As the variance increases, the degrees of freedom decrease, which can impact the accuracy of the results. However, it is important to note that a larger sample size can often lead to a more accurate estimate of the population variance, even if the degrees of freedom are not directly related to the sample size.
When calculating the sample variance, the degrees of freedom is equal to the sample size (n) minus 1, often denoted as (n-1). This is because we lose one degree of freedom when estimating the population means using the sample mean.
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The cumulative distribution function of random variable V is f_V (v) = {0, v < -5 (v + 1)^2/144, -5 lessthanorequalto v < 7 1, x greaterthanorequalto 7 What are the expected value and variance of V? What is E[V^3]?
The expected value of V is 1.75
The variance of V is 97.65
E[V³] is 193.083
To find the expected value and variance of V, we first need to find the distribution function of V. For -5 ≤ v < 7, the cumulative distribution function (CDF) F_V(v) can be found by integrating f_V(v):
F_V(v) = ∫ f_V(t) dt
= ∫ (t+1)²/144 dt
= (1/144) * ∫ (t² + 2t + 1) dt
= (1/144) * [(t³)/3 + t² + t]_(-5)^(v)
= (1/144) * [(v³)/3 + v² + v + 160]/3
For v ≥ 7, F_V(v) = 1.
V's expected value is:
E[V] = ∫ v f_V(v) dv = ∫ (-∞ to -5) 0 dv + ∫ (-5 to 7) v*(v+1)²/144 dv + ∫ (7 to ∞) v dv
= (1/144) * ∫ (-5 to 7) (v³ + v²) dv + ∫ (7 to ∞) v dv
= (1/144) * [(7⁴ - (-5)⁴)/4 + (7³ - (-5)³)/3 + 7²*(7-(-5))]
= 1.75
V's variance is as follows:
Var[V] = E[V²] - (E[V])²
= ∫ v² f_V(v) dv - (E[V])²
= ∫ (-∞ to -5) 0 dv + ∫ (-5 to 7) v²*(v+1)²/144 dv + ∫ (7 to ∞) v² dv - (E[V])²
= (1/144) * ∫ (-5 to 7) (v⁴ + 2v³ + v²) dv + ∫ (7 to ∞) v² dv - (E[V])²
= (1/144) * [(7⁵ - (-5)⁵)/5 + 2*(7⁴ - (-5)⁴)/4 + (7³3 - (-5)³)/3 + 7*(7²*(7-(-5))) - (1.75)²]
= 97.65
Finally, we can find E[V³] using:
E[V³] = ∫ v³ f_V(v) dv
= ∫ (-∞ to -5) 0 dv + ∫ (-5 to 7) v³*(v+1)²/144 dv + ∫ (7 to ∞) v³ dv
= (1/144) * ∫ (-5 to 7) (v⁵ + 2v⁴ + v³) dv + ∫ (7 to ∞) v³ dv
= (1/144) * [(7⁶ - (-5)⁶)/6 + 2*(7⁵ - (-5)⁵)/5 + (7⁴ - (-5)⁴)/4 + 7*(7³ - (-5)³)/3]
= 193.083
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A cylindrical tank, lying on its side, has a radius of 10 ft^2 and length 40ft. Assume that the tank is half-filled with water and that water leaks through a hole in the bottom of area B = 1 in^2. Determine the water level y(t) and the time te when the tank is empty. y(t) = te = seconds.
The water level y(t) = √(1000 - 80πt/3), te ≈ 11.8 seconds.
The water level y(t) in the cylindrical tank with radius 10 ft and length 40 ft decreases over time until the tank is empty at time t=te seconds can be found shown below:
First, find the volume of the half-filled tank: V = (1/2)π(10^2)(40) = 2000π ft³. The leakage rate Q = (1 in²)(1/144 ft²/in²) = 1/144 ft². Since Q = dV/dt, we have dV = -Qdy.
Integrating both sides gives V = -Qy + C. Initially, V = 2000π and y = 10, so C = 3000π. Thus, V = -Qy + 3000π. Solving for y, we get y(t) = √(1000 - 80πt/3). To find te, set V = 0 and solve for t: 0 = -80πt/3 + 1000, which gives te ≈ 11.8 seconds.
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In a certain year, according to a national Census Bureau, the number of people in a household had a mean of 4.664.66 and a standard deviation of 1.941.94.
This is based on census information for the population. Suppose the Census Bureau instead had estimated this mean using a random sample of 225 homes. Suppose the sample had a sample mean of 4.8 and standard deviation of 2.1
Describe the center and variability of the data distribution. what would you predict as the shape of the data distribution? explain. The center of the data distribution is ______.
The variability of the population distribution is _____.
It's reasonable to assume the sample distribution's shape would be similar to the population distribution's shape. However, without more information, we cannot confirm the exact shape of the distribution.
The center of the data distribution is represented by the mean. According to the national Census Bureau, the mean number of people in a household for the entire population is 4.66.
The variability of the population distribution is represented by the standard deviation. In this case, the standard deviation provided by the Census Bureau is 1.94.
So, the center of the data distribution is 4.66, and the variability of the population distribution is 1.94.
Since the Census Bureau has used a random sample of 225 homes, the sample mean (4.8) and standard deviation (2.1) could be used to estimate the population mean and standard deviation. However, these sample statistics are not necessarily equal to the population parameters.
As for the shape of the data distribution, it's difficult to predict without more information about the distribution itself. If the data is normally distributed, the shape would be bell-shaped. If the sample is representative of the population, it's reasonable to assume the sample distribution's shape would be similar to the population distribution's shape. However, without more information, we cannot confirm the exact shape of the distribution.
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In AABC, mA = 70° and m28=35".
Select the triangle that is similar to AABC.
A. APQR, in which m2P = 70° and
mAR= 75°
B. AMNP, in which mM= 70° and
m2N = 105
C. AJKL, in which mJ = 35° and
mZL=105"
D. ADEF in which m2D = 75° and
mZF=15°
Note that where in triangle ABC, m∠A = 70° and m∠8=35" the dimension that are similar to the above is: Option A ΔPQR, in which m∠P = 70° and m∠R= 75°
How is this so?Note that for the triangles to be similar, they must have the same internal angles or angles in a similar ratio.
We know that the angles 70° and 35°. By subtracting these from ΔABC we get the third angle which is ∠75°
So since to be a similar triangle, they must have the same angles, note that he only triangle with similar properties is ΔPQR because:
m∠P = 70° and m∠R= 75°.
180 - (70+75) = 35°
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help me out pls!!! :)
Answer: 254.34
Step-by-step explanation:
First we can fine the radius by dividing the diameter (18in) by 2: 18/2=9
Then we can use the formula to find the area of the circle (pi*r^2):9*9*pi=81pi
Finally, approximate pi to 3.14 and multiple: 81*3.14=254.34
Therefore, the answer is 254.34
Answer:1017.36
Step-by-step explanation:
a bed cost $1500 cash, but on hired purchase one must pay $250 down payment and $140 every month for a year. how much interest does one pay if one were to buy the bed on hire purchased?
The amount of interest paid if one were to buy the bed on hire purchased is $430
How much interest does one pay if one were to buy the bed on hire purchased?Cost of the bed = $1500
Down payment = $250
Monthly payment = $140
Number of months = 12
Total payment made on hired purchase = Down payment + (Monthly payment × Number of months)
= 250 + (140 × 12)
= 250 + 1,680
= $1,930
Amount of interest paid = Total payment made on hired purchase - Cost of the bed
= $1,930 - $1,500
= $430
In conclusion, the total interest paid on hired purchase is $430
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3. The length of one side of a right triangle is shown in this diagram. What could be the lengths of the two remaining sides of the triangle?
A. 24cm and 26 cm
B. 13 cm and 24 cm
C. 7 cm and 14 cm
D. 12 cm and 22 cm
Answer:
A. 24cm and 26cm
Step-by-step explanation:
Pythagorean theorem.
A^2+B^2=C^2
10^2+24^2=26^2
The other options plugged into this formula would make a false statement.
use formula for arc length to show that the circumference of a circle x^2+y^2=1 is 2pi
The circumference of the circle x² + y² = 1 is 2π.
To show that the circumference of the circle x² + y² = 1 is 2π, we can use the arc length formula. The formula for arc length (s) in a circle is given by:
s = r × θ
where r is the radius of the circle and θ is the central angle in radians.
For the circle x² + y² = 1, the radius (r) is equal to 1 (since the equation is already in the standard form). To find the circumference, we need to find the arc length for a complete circle. A complete circle has a central angle of 2π radians. Therefore, we can plug these values into the arc length formula:
Circumference = s = r × θ
Circumference = 1 × 2π
Circumference = 2π
Thus, the circumference of the circle x² + y² = 1 is 2π.
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Answer:
2
Step-by-step explanation:
determine the sample size n needed to construct a 90onfidence interval to estimate the population mean when = 36and the margin of error equals .4
You need a sample size of approximately 10,875 to construct a 90% confidence interval with a margin of error of 0.4 and a standard deviation of 36.
To determine the sample size (n) needed to construct a 90% confidence interval for estimating the population mean, given a standard deviation (σ) of 36 and a margin of error of 0.4, you can use the formula:
[tex]n = (Z * σ / E)^2[/tex]
where:
n = sample size
Z = Z-score corresponding to the desired confidence level (90%)
σ = standard deviation (36)
E = margin of error (0.4)
For a 90% confidence interval, the Z-score is 1.645. Now, plug in the values into the formula:
n = (1.645 * 36 / 0.4)^2
n ≈ 10874.09
Since sample size should be a whole number, round up to the nearest whole number: n ≈ 10875.
So, you need a sample size of approximately 10,875 to construct a 90% confidence interval with a margin of error of 0.4 and a standard deviation of 36.
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An investor has an account with stock from two different companies. Last year, her stock in Company A was worth $5750 and her stock in Company B was worth $1200. The stock in Company A has decreased 16% since last year and the stock in Company B has decreased 2%. What was the total percentage decrease in the investor's stock account? Round your answer to the nearest tenth (if necessary).
The total percentage decrease in the investor's stock account is 13.6%
What is percentage?Percentage is a way of expressing a number as a fraction of 100. It is denoted using the symbol "%". For example, 25% is the same as 25/100 or 0.25.
According to given information:To find the total percentage decrease in the investor's stock account, we need to first calculate the new values of the stocks after the decreases and then find the percentage decrease of the total value compared to the original value.
The new value of the stock in Company A is:
5750 - 0.16 * 5750 = 4830
The new value of the stock in Company B is:
1200 - 0.02 * 1200 = 1176
The total value of the stocks after the decreases is:
4830 + 1176 = 6006
The percentage decrease of the total value compared to the original value is:
(1 - 6006/6950) * 100% = 13.6%
Therefore, the total percentage decrease in the investor's stock account is 13.6% (rounded to the nearest tenth).
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Can someone help me with this pleaseeee.
Find all the sides and angles of the triangle
Step-by-step explanation:
first, the law of cosine (the rule of Pythagoras generalized for any type of triangle) :
c² = a² + b² - 2ab×cos(C)
c is the side opposite of the angle C, a and b are the other 2 sides.
in our case :
b² = 5² + 8² - 2×5×8×cos(51)
b² = 25 + 64 - 80×cos(51) =
= 89 - 80×cos(51) = 38.65436872...
b = 6.217263764... ≈ 6.22
now we have all 3 sides and need to find the other 2 angles.
law of sine
a/sin(A) = b/sin(B) = c/sin(C)
a, b, c are the sides, and A, B, C are the corresponding opposite angles.
5/sin(A) = 6.217263764.../sin(51)
sin(A) = 5×sin(51)/6.217263764... =
= 0.624990342...
A = 38.6814786...° ≈ 38.68°
sin(C) = 8×sin(51)/6.217263764... =
= 0.999984547...
C = 89.6814786...° ≈ 89.68°
the switch has been in its starting position for a long time before moving at t = 0. find v(t), i1(t), and i2(t) for t > 0 .
When the switch changes position at t = 0, the circuit will be in a transient state until it reaches a steady state. Let's analyze the circuit in both the transient and steady state.
Transient State (t < 0):
Since the switch has been in its starting position for a long time, the circuit has reached a steady state, which means that all voltages and currents are constant. Therefore, we can assume that v(t<0) = V0, i1(t<0) = 0, and i2(t<0) = 0.
Steady State (t ≥ 0):
When the switch changes position at t = 0, the voltage source is connected to the resistors R1 and R2 in series. Therefore, the voltage across R1 and R2 is equal to V0.
The current flowing through the resistors is given by Ohm's law:
i = V/R
where i is the current, V is the voltage, and R is the resistance.
Using this equation, we can find the current flowing through R1 and R2:
i1(t) = V0 / R1
i2(t) = V0 / R2
Since the circuit is a series circuit, the current flowing through the circuit is the same as the current flowing through R1 and R2. Therefore,
i(t) = i1(t) = i2(t) = V0 / (R1 + R2)
The voltage across R1 is given by:
v(t) = i1(t) * R1 = V0 * R2 / (R1 + R2)
Therefore, the solutions for v(t), i1(t), and i2(t) for t ≥ 0 are:
v(t) = V0 * R2 / (R1 + R2)
i1(t) = V0 / R1
i2(t) = V0 / R2
i(t) = V0 / (R1 + R2)
where V0 is the voltage of the voltage source, R1 and R2 are the resistances of resistors R1 and R2, respectively.
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similar to 3.10.1 in rogawski/adams. how fast is the water level rising if water is filling a rectangular bathtub with a base of 28 square feet at a rate of 5 cubic feet per minute? rate is =
The water level is rising at a rate of 0.006 feet per minute. This can be answered by the concept of Differentiation.
The formula for the volume of a rectangular box is V = lwh, where l, w, and h represent the length, width, and height respectively. Since the base of the bathtub is 28 square feet, we can assume that the length and width are both 28 feet. Let's say the height of the water in the bathtub is h at time t.
We know that the water is filling the bathtub at a rate of 5 cubic feet per minute, so the rate of change of the volume of water in the bathtub is 5. We want to find the rate of change of the height of the water, which we can call dh/dt.
Using the formula for the volume of a rectangular box, we can write:
V = lwh = 28wh
We can differentiate both sides with respect to time t:
dV/dt = 28w dh/dt
We know that dV/dt is 5, and w is also 28 since the base of the bathtub is a rectangle with sides of length 28 feet. Therefore, we can solve for dh/dt:
5 = 28(28) dh/dt
dh/dt = 5/(28×28)
dh/dt = 0.006 ft/min
Therefore, the water level is rising at a rate of 0.006 feet per minute.
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Once a week, Ms. Conrad selects one student at random from her class list to win a “no homework”
pass. There are 17 girls and 18 boys in the class. Rounded to the nearest percent, what is the
probability that a girl will win two weeks in a row?
The probability that a girl will win two weeks in a row is 24%.
What is probability?
Probability tells how many times something will happen or be present.
The probability of a girl winning in a given week is 17/35 since there are 17 girls and 35 students total. Assuming each week's selection is independent of previous selections, the probability of a girl winning two weeks in a row is (17/35) x (17/35) = 289/1225.
Rounding this to the nearest percent gives a probability of 24%.
Therefore, the probability is 24%.
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find the least squares regression line for the points. (0, 0), (2, 2), (3, 6), (4, 7), (5, 9)
Answer:
Use the graphing calculator to plot the points and then generate the least squares regression line.
y = 1.87837878 - .4594594595
What are the values of AB and DE in parallelogram ABCD? AB= (Type an integer or a decimal.) B A 22 17 с 11 E D ** Q G
The values of AB and DE in parallelogram ABCD are AB = 14 and DE = 5
What are the values of AB and DE in parallelogram ABCD?From the question, we have the following parameters that can be used in our computation:
The parallelogram ABCD
By the properties of a parallelogram;
The opposite sides of a parallelogram are congruent
This means that
AB = CD = 14
AE + DE = BC
So, we have
19 + DE = 24
Evaluate
DE = 5
Hence, the value of DE is 5 units
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Select true or false.
(a) T/F 22 ≡ 8 (mod 7)
(b) T/F −13 ≡ 9 (mod 12)
(c) T/F −13 ≡ 9 (mod 11)
(d) T/F −13 ≡ 12 (mod 2)
Hence, a and c are true .
What is the modulo?When two numbers are split, the modulo operation in computing the remainder of the division. A modulo n is the remainder of the Euclidean division of two positive numbers, a and n, where a is the dividend and n is the divisor.
What is the congruent modulo ?A congruence relation is an equivalence relation that is symbol of addition, subtraction, and multiplication. Congruence modulo n is one one map connection. The symbol for congruence modulo n is: The brackets indicate that (mod n) applies to both sides of the equation, not only the right-hand side of the equation.
True,because they are congruent modulo 7 because 22 divided by 7 leaves a remainder of 1, and the divided of 8 by 7 leaves a remainder of 1.False because they are not congruent modulo 12 because 13 divided by 12 leaves a remainder of 1, and 9 divided by 12 leaves a remainder of 9.True because they are congruent modulo 11 and 13 divided by 11 leaves a remainder of 2 and 9 divided by 11 leaves a remainder of 9.False because they are not congruent modulo 2 since 13 divided by 2 leaves a remainder of 1, and 12 divided by 2 leaves a remainder of 0.Learn more about mod here:
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Use the Maclaurin series for cos(x) to compute cos(3) correct to five decimal places. (Round your answer to five decimal places.) 0.99862
Maclaurin series for cos(x) to compute [tex]\cos(3) \approx 0.99862$.[/tex]
What is Maclaurin series?
The Maclaurin series is a special case of the Taylor series, which is a power series expansion of a function about 0. The Maclaurin series is obtained by setting the center of the Taylor series to 0. It is named after the Scottish mathematician Colin Maclaurin.
The Maclaurin series of a function f(x) is given by:
[tex]f(x) = f(0) + f'(0)x + (f''(0)/2!)x^2 + (f'''(0)/3!)x^3 + ... + (f^{(n)}(0)/n!)x^n + ...[/tex]
where [tex]f^{(n)}(0)[/tex] denotes the nth derivative of f evaluated at 0.
Using the Maclaurin series for [tex]$\cos(x)$[/tex], we have:
[tex]\cos(x) &= \sum_{n=0}^\infty \frac{(-1)^n}{(2n)!}(x)^{2n}[/tex]
Substituting [tex]$x=3$[/tex] into this series, we get:
[tex]\cos(3) &= \sum_{n=0}^\infty \frac{(-1)^n}{(2n)!}(3)^{2n}[/tex]
[tex]&= 1 - \frac{3^2}{2!} + \frac{3^4}{4!} - \frac{3^6}{6!} + \frac{3^8}{8!} - \cdots[/tex]
[tex]&\approx 0.99862 \quad\text{(correct to five decimal places)}[/tex]
Therefore, [tex]\cos(3) \approx 0.99862$.[/tex]
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Graph the function f(x) = 32. Plot the key features including any x- and y-intercepts, any vertical, horizontal, or slant asymptotes, and any holes.
The graph of the function f(x) = 32 is attached accordingly.
How would you describe the above graph ?X - Intercept - There is no x-intercept since the function is a horizontal line.
Y -Interept - The y-intercept is (0, 32), since the line intersects the y-axis at y = 32.
Vertical Asymptotes - There are 0 vertical asymptotes, since t function is defined for all values of x.
Horizontal Asymptotes - There are 0 horizontal asymptotes, since the function is a horizontal line.
Slant Asymptotes - There are zeroslant asymptotes, since the function is a horizontal line.
Holes - There are zeroholes in the graph, since the function is a horizontal line with no breaks or discontinuities.
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Answer: see below
Step-by-step explanation:
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suppose that initially c = 2 0.75 × gdp, i = 3, g = 2, and nx = 1. compute the equilibrium value of spending.
The equilibrium value of spending is 72.
To compute the equilibrium value of spending, we need to use the equation for the expenditure approach to GDP:
GDP = C + I + G + NX
Where:
C = consumption
I = investment
G = government spending
NX = net exports
Given the values of c, i, g, and nx, we can substitute them into the equation:
GDP = 2.75 × GDP + 3 + 2 + 1
Simplifying the equation, we get:
GDP = 2.75 × GDP + 6
Now, we can solve for GDP:
GDP - 2.75 × GDP = 6
0.25 × GDP = 6
GDP = 24
Therefore, the equilibrium value of spending is:
C + I + G + NX = 2.75 × GDP + 3 + 2 + 1 = 2.75 × 24 + 6 = 72
The equilibrium value of spending is 72.
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There are 23 rabbits in a valley. The rabbit population grows at a rate of approximately 18% per month. The approximate number of rabbits in the valley after n months is given by this formula: number of rabbits - 23 × 1.18n Use this formula to predict the number of rabbits in the valley after 25 months. Round your answer to the nearest integer.
Evaluating the exponential equation we can see that after 25 months there will be 1,441 rabbits after 25 months.
How to find the number of rabbits in the valley after 25 months?We know that the population of rabbits is modeled by the exponential equation below:
P(n) = 23*1.18^n
Where n is the number of months.
Then the population after 25 months is what we get when we evaluate the exponential equation in n = 25, we will get:
P(25) = 23*1.18^25 = 1,441
There will be 1,441 rabbits after 25 months.
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At a certain university, the probability that an entering freshman will graduate in 4 years is .65. If in the incoming class of 2017, there were 1025 freshman, determine the following probabilities.Exactly 697 will graduate in 4 years.At most 685 will graduate in 4 years.650 or more will graduate in 4 years.Between 665 and 715 (inclusive) will graduate in 4 years.
Let X be the number of students who will graduate in 4 years out of 1025 students. The final answer of probabilities is [tex]P(X = 697)[/tex][tex]=0.080[/tex]; [tex]P(X \leq 685) = 0.123[/tex][tex]P(X \geq 650) = 0.997[/tex][tex]P(665 \leq X \leq 715) =0.826[/tex]
Then X follows a binomial distribution for finding the probabilities with n = 1025 and p = 0.65.
(a) [tex]P(X = 697) = (1025 choose 697) * (0.65)^697 * (1-0.65)^(1025-697)[/tex]=[tex]0.080[/tex]
(b) [tex]P(X ≤ 685)[/tex]= [tex]Σ_(k=0)^685 (1025 choose k) * (0.65)^k * (1-0.65)^(1025-k)[/tex]= [tex]0.123[/tex]
(c) [tex]P(X ≥ 650)[/tex] = [tex]1 - P(X < 650) = 1 - Σ_(k=0)^649 (1025 choose k) * (0.65)^k * (1-0.65)^(1025-k)[/tex]= [tex]0.997[/tex]
(d) [tex]P(665 ≤ X ≤ 715)[/tex]= [tex]Σ_(k=665)^715 (1025 choose k) * (0.65)^k * (1-0.65)^(1025-k)[/tex]≈[tex]0.826[/tex]
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