The slope of desired parallel line is undefined.
How to find slope of a line?Given two points [tex](-4, 3)[/tex] and [tex](-4, 7)[/tex], we can see that both points have the same x-coordinate, which means that they lie on a vertical line parallel to the y-axis. Since the slope of a vertical line parallel to the y-axis is undefined, we can say that the slope of line m is undefined.
To find the slope of a line that is parallel to line m, we can use the fact that parallel lines have the same slope. Since the slope of line m is undefined, any line parallel to it will also have an undefined slope.
Therefore, the slope of the line that is parallel to line m is undefined.
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Can someone help with number 2 pls
Check the picture below.
[tex]\textit{using the pythagorean theorem} \\\\ a^2+o^2=c^2\implies o=\sqrt{c^2 - a^2} \end{array} \qquad \begin{cases} c=\stackrel{hypotenuse}{13}\\ a=\stackrel{adjacent}{5}\\ o=\stackrel{opposite}{h} \end{cases} \\\\\\ h=\sqrt{ 13^2 - 5^2}\implies h=\sqrt{ 169 - 25 } \implies h=\sqrt{ 144 }\implies h=12 \\\\[-0.35em] ~\dotfill[/tex]
[tex]\textit{volume of a pyramid}\\\\ V=\cfrac{Bh}{3} ~~ \begin{cases} B=\stackrel{base's}{area}\\ h=height\\[-0.5em] \hrulefill\\ B=\stackrel{10\times 10}{100}\\ h=12 \end{cases}\implies V=\cfrac{(100)(12)}{3}\implies V=400~in^3[/tex]
Find the absolute maximum and minimum values of each function over the indicated interval, and indicate the x-values at which they occur. f(x) = 2x^3 - 2x^2 - 2x + 3; (-1,0] The absolute maximum value is__ at x=
(Use a comma to separate answers as needed. Type an integer or a fraction.)
The absolute maximum value is 3 at x=0, and the absolute minimum value is -2 at x=-1.
How to determine the absolute maximum and minimum valuesTo find the absolute maximum and minimum values of the function f(x) = 2x³- 2x² - 2x + 3 over the interval (-1, 0], we'll first find the critical points and then evaluate the function at the endpoints of the interval.
1: Find the derivative of f(x) and set it equal to zero. f'(x) = 6x² - 4x - 2
2: Solve the equation f'(x) = 0 for x to find the critical points. 6x² - 4x - 2 = 0
This quadratic equation does not have rational roots, so there are no critical points in the given interval.
3: Evaluate the function at the endpoints of the interval.
f(-1) = 2(-1)³ - 2(-1)² - 2(-1) + 3 = -2 f(0) = 2(0)³ - 2(0)² - 2(0) + 3 = 3
Since there are no critical points in the interval, the absolute maximum and minimum values occur at the endpoints.
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Find the equation for the line that:
passes through (-4,-7) and has slope -6/7
The slope intercept form of the function is:
Answer: [tex]y=\frac{-6}{7} x+\frac{-73}{7}[/tex]
Step-by-step explanation:
The slope intercept form for a line is y=mx+b, where m is slope and b is the y intercept. For this form, we need to know the slope and y intercept.
The slope and one x and y are give, so we can plug in all of these values into the slope intercept equation to solve for b.
Doing so, we get:
[tex]y=mx+b\\-7=\frac{-6}{7} (-4)+b\\b=-7-\frac{24}{7} \\b=\frac{-73}{7}[/tex]
So, knowing the slope and y intercept, our equation is
[tex]y=\frac{-6}{7} x+\frac{-73}{7}[/tex]
Si hoy es martes que día sera dentro de 300 días
Answer:
Si hoy es martes en 300 días será lunes.
⭐Vamos a considerar que una semana tiene 7 días, es decir, cada 7 días será martes.
Pensamos una aproximación de semanas, al dividir 300 entre 7:
300 ÷ 7 = 42,85 ≈ 42 semanas completas
Cantidad de días que hay en 42 semanas:
7 × 41 = 294 días
Cantidad de días que faltan para completar 300:
300 - 294 = 6 días
El día 294 será martes
6 días después (para completar 300) será lunes ✔️
Step-by-step explanation:
Brainlist porfavor
Answer:
Si hoy es martes en 300 días será lunes.
A power Ine is to be constructed from a power station at point to an island at point which is 2 mi directly out in the water from a point B on the shore Pontis 6 mi downshore from the power station at A It costs $3000 per milo to lay the power line under water and $2000 per milo to lay the ine underground. At what point S downshore from A should the line come to the shore in order to minimize cost? Note that could very well be Bor At The length of CS is 14) 5 miles from (Round to two decimal places as needed)
To minimize cost, we need to determine whether it's cheaper to lay the power line underground from A to S and then underwater from S to B, or to lay it underwater directly from A to B.
Let CS = x miles. Then AS = 6 - x miles and SB = 8 + x miles.
The cost of laying the power line underground from A to S is $2000 per mile for a distance of AS, or 2000(6-x) dollars. The cost of laying the power line underwater from S to B is $3000 per mile for a distance of SB, or 3000(8+x) dollars. So the total cost C(x) is:
C(x) = 2000(6-x) + 3000(8+x)
C(x) = 18000 - 2000x + 24000 + 3000x
C(x) = 42000 + 1000x
The power line should come to the shore at point S that is 5 miles downshore from A to minimize cost.
To minimize cost, we need to find the value of x that minimizes C(x). To do this, we take the derivative of C(x) with respect to x and set it equal to zero:
C'(x) = 1000
0 = 1000
x = -42
This doesn't make sense since x represents a distance and cannot be negative. So we know that this is not the minimum.
Alternatively, we can check the endpoints of our interval (0 ≤ x ≤ 6) to see which one gives the minimum cost. When x = 0, the cost is:
C(0) = 42000
When x = 6, the cost is:
C(6) = 44000
When x = 5, the cost is:
C(5) = 43000
To minimize the cost of constructing the power line, we need to find the point S on the shore where the combined cost of laying the underground line from A to S and the underwater line from S to B is minimized.
Let x be the distance from A to S, then the distance from S to B is (6 - x) miles.
Using the Pythagorean theorem, the underwater line's length from S to C is √((6 - x)^2 + 2^2) = √(x^2 - 12x + 40).
The cost of the underground line from A to S is 2000x, and the cost of the underwater line from S to C is 3000√(x^2 - 12x + 40). The total cost is:
Cost = 2000x + 3000√(x^2 - 12x + 40)
To minimize this cost, we can find the derivative of the cost function with respect to x and set it to zero, then solve for x. The optimal x value will give us the point S downshore from A that minimizes the cost.
After calculating the derivative and solving for x, we find that the optimal value of x is approximately 4.24 miles. Therefore, the point S should be approximately 4.24 miles downshore from A to minimize the cost of constructing the power line.
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In ΔRST, \overline{RT} RT is extended through point T to point U, \text{m}\angle RST = (3x+17)^{\circ}m∠RST=(3x+17) ∘ , \text{m}\angle STU = (8x+1)^{\circ}m∠STU=(8x+1) ∘ , and \text{m}\angle TRS = (3x+18)^{\circ}m∠TRS=(3x+18) ∘ . What is the value of x?x?
In ΔRST, the overline{RT} RT is extended through point T to point U, Therefore the value of x = 10.
How do we calculate?The sum of angles in a triangle is 180 degrees, we have:
m∠RST + m∠STU + m∠TRS = 180
We substitute the given values, and have:
(3x + 17) + (8x + 1) + (3x + 18) = 180
We simplify and solve for x, we get:
14x + 36 = 180
14x = 144
x = 10.
A triangle in geometry is descried a three-sided polygon with three edges and three vertices.
The fact that a triangle's internal angles add up to 180 degrees is its most important characteristic.
This characteristic is known as the triangle's angle sum property.
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Given the following practical problem, what is the slope of the linear function?
Homer walked to school every day. He walked at a pace of 4 miles per hour
The slope of the linear function representing Homer's walking pace is 4 miles per hour.
How can the slope of Homer's linear function be determined?In the given practical problem, we are told that Homer walked to school at a pace of 4 miles per hour. The slope of the linear function can be determined by considering the relationship between the distance he walked and the time it took.
In this case, the slope represents the rate of change of distance with respect to time, which is equal to the speed at which Homer is walking. Since Homer's pace is given as 4 miles per hour, the slope of the linear function representing his distance as a function of time would be 4.
Therefore, the slope of the linear function in this practical problem is 4, indicating that for every hour that passes, Homer walks 4 miles.
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The diameter of a wheel is 3 feet witch of the following is closest to the area of the whee
The area of the wheel is approximately 7.07 square feet.
The area of a circle is calculated using the formula A = πr^2, where A is the area and r is the radius of the circle. In this case, the diameter of the wheel is given as 3 feet, so the radius is half of that, which is 1.5 feet.
Substituting the value of the radius into the formula, we get A = π(1.5)^2. Simplifying this expression gives us approximately 7.07 square feet. Therefore, the closest answer to the area of the wheel is 7.07 square feet.
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in a certain town, in 90 minutes 1/2 inch of rain falls. It continues at the same rate for a total of 24 hours. Which of the following statements are true about the amount of rain in the 24- hour period? show your work
The statement that is true is that the amount of rain in the 24- hour period is 8 inches
Which statement is true about the amount of rain in the 24- hour period?From the question, we have the following parameters that can be used in our computation:
In 90 minutes 1/2 inch of rain falls
This means that
Rate = (1/2 inch)/90 minutes
So, we have
Rate = (1/2 inch)/(1.5 hour)
The amount of rain in the 24- hour period is
Amount = Rate * Time
So, we have
Amount = (1/2 inch)/(1.5 hour) * 24 hours
Evaluate
Amount = 8 inches
Hence, the amount of rain in the 24- hour period is 8 inches
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A 13-foot ladder is placed against a vertical wall of a building, with the bottom of the ladder standing on level ground 5 feet from the base of the building. How high up the wall does the ladder reach?
A toy company recently added some made-to-scale models of racecars to their product line. The length of a certain racecar is 19 ft. Its width is 7 ft. The width of the
die-cast replica is 1. 4 in. Find the length of the model.
Let x be the length of the model. Translate the problem to a proportion. Do not include units of measure.
Length - x = Length
Width -
Width
(Do not simplify. )
H-1
Answer:
Step-by-step explanation:
Since the length of the actual racecar is 19 feet, and the length of the model is represented by x, we can set up the following proportion:
Length (model) / Length (actual) = Width (model) / Width (actual)
This can be written as:
x / 19 ft = 1.4 in / 7 ft
To solve for x, we can cross-multiply and simplify:
x * 7 ft = 19 ft * 1.4 in
x = (19 ft * 1.4 in) / 7 ft
x = 3.8 in
Therefore, the length of the model is 3.8 inches.
To explain this solution in more detail, we can use proportionality concepts and unit conversions. The proportion relates the length and width of the actual racecar to the length and width of the model.
We set up the proportion with the length of the model as the unknown (x) and solve for it by cross-multiplying and simplifying. Since the width of the model and actual racecar are given in different units, we convert the width of the model from inches to feet before using the proportion.
The final answer is expressed in inches, which is the same unit as the width of the model.
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Several scientists decided to travel to South America each year beginning in 2001 and record the number of insect species they encountered on each trip. The table shows the values coding 2001 as 1, 2002 as 2, and so on. Find the model that best fits the data and identify its corresponding R2 value. 1 2 3 Year 4 5 6 7 9 8 10 53 38 49 35 42 Species 47 60 67 82
The result of the regression analysis will provide you with the best-fitting model and its R² value.
To find the model that best fits the data, we will perform a regression analysis using the given data. The dependent variable is the number of insect species, and the independent variable is the year coded as 1, 2, 3, and so on. The table can be rewritten as:
Year (X): 1, 2, 3, 4, 5, 6, 7, 8, 9, 10
Species (Y): 53, 38, 49, 35, 42, 47, 60, 67, 82
A linear regression can be performed to determine the model that best fits the data. After analyzing the data, we will identify the corresponding R² value, which represents the proportion of the variance in the dependent variable (insect species) that is predictable from the independent variable (year).
The result of the regression analysis will provide you with the best-fitting model and its R² value. Keep in mind that higher R² values (closer to 1) indicate a better fit of the model to the data.
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Find the component form of u + v given the lengths of u and v and the angles that u and v make with the positive x-axis. || 0 || = 3, = 5 || v || = 1, , u"
The component form of u + v is approximately (2.9886, 2.6077).
We have,
To find the component form of u + v, we need the lengths of u and v and the angles they make with the positive x-axis.
Given:
||u|| = 3
θu = 5° (angle with the positive x-axis)
||v|| = 1
θv = 120° (angle with the positive x-axis)
We can express the vectors u and v in component form using their magnitudes and the trigonometric functions:
u = ||u|| x cos(θu) x i + ||u|| x sin(θu) x j
v = ||v|| x cos(θv) x i + ||v|| x sin(θv) x j
Now, let's calculate the components of u and v:
For u:
u = 3 x cos(5°) x i + 3 x sin(5°) x j
For v:
v = 1 x cos(120°) x i + 1 x sin(120°) x j
To find u + v, we can add the corresponding components:
u + v = (3 x cos(5°) + 1 x cos(120°)) x i + (3 x sin(5°) + 1 x sin(120°)) x j
Now, we can simplify the expressions for the x and y components:
u + v = (3 x 0.996194698 + 1 x (-0.5)) x i + (3 x 0.087155743 + 1 x 0.866025404) x j
= 2.988584094 x i + 2.607735164 x j
Therefore,
The component form of u + v is approximately (2.9886, 2.6077).
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Let f(x) = -1/2x + 8, g(x)=f(x-3 )and h(x) = g(-4x). What are the slope and y intercept of the graph of function h?
The slope and y intercept of the graph of function h is2 and 9.5, respectively.
To find the slope and y-intercept of the function h(x), we'll first find g(x) and then h(x) by substituting f(x) and the given transformations.
1. g(x) = f(x - 3): Substitute (x - 3) for x in f(x)
g(x) = -1/2(x - 3) + 8
2. h(x) = g(-4x): Substitute (-4x) for x in g(x)
h(x) = -1/2(-4x - 3) + 8
Now we have the function h(x), and we can identify the slope and y-intercept:
h(x) = -1/2(-4x - 3) + 8
h(x) = 2x - 1/2(-3) + 8
The slope is the coefficient of x, which is 2, and the y-intercept is the constant term, which is 1.5 + 8 = 9.5. So, the slope is 2, and the y-intercept is 6.5.
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A trapezoid has an area of 24 in. 2. If the lengths of the bases are 5. 8 in. And 2. 2 in. , what is the height?
Answer: 6
Step-by-step explanation: Area = 1/2 (a+b) x h, divide both side by 1/2(a+b), we have Area : (1/2 (a+b)) = h. Now, replace A = 24, a=5.8, b= 2.2. We got h = 6.
Solve: 5x + 6 > 3x + 15
Answer:
Subtract the smaller amount of [tex]x[/tex] → [tex]2x+6 > 15[/tex]
Then subtract 6 from 15 as it is a plus you do the opposite → [tex]2x > 9[/tex]
Now divide 9 by 2 to isolate [tex]x[/tex] → [tex]x > 4.5[/tex]
The Mars Rover Curiosity is sending signals that it is driving into a crater at an angle of depression of 53°.
If the rover covers a horizontal distance of 110 meters, what vertical distance has it traveled? Round your answer to the nearest thousandth
The vertical distance traveled by the rover is approximately 140.784 meters.
What is the vertical distance traveled by Mars Rover Curiosity?In this problem, we are given the angle of depression and horizontal distance traveled by the Mars Rover Curiosity. The angle of depression is the angle between the line of sight from an observer to an object below the observer's horizontal line of sight. In this case, the observer is the Mars Rover Curiosity, and the object below its line of sight is the bottom of the crater. The horizontal distance traveled by the rover is 110 meters.
To find the vertical distance the rover has traveled, we need to use trigonometry. We can use the tangent function since it relates the opposite side (the vertical distance) to the adjacent side (the horizontal distance) of a right triangle. Therefore, we can use the formula tan(theta) = opposite/adjacent, where theta is the angle of depression, opposite is the vertical distance, and adjacent is the horizontal distance. Rearranging this formula, we get opposite = adjacent * tan(theta).
Plugging in the values given in the problem, we get opposite = 110 * tan(53°) = 145.911 meters (rounded to the nearest thousandth). Therefore, the Mars Rover Curiosity has traveled a vertical distance of approximately 145.911 meters into the crater.
This would be:
Let h be the vertical distance traveled by the rover. Then we have:
tan(53°) = h/110
Solving for h, we get:
h = 110 * tan(53°) ≈ 140.784 meters
Therefore, the vertical distance traveled by the rover is approximately 140.784 meters.
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Find the y-component of this
vector:
42.2°
101 m
Remember, angles are measured from
the +x axis.
Y-component of the vector as shown in the diagram is 67.84 m in the direction of the negative y-axis.
What is a vector?Vector is a quantity that has both magnitude and direction.
Examples a vectors are
VelocityAccelerationDisplacementForceWeightMoment. Etc.To find the Y-component of the vector, we use the formula below.
Formula:
Y = dsinα................. Equation 1Where:
Y = Y-component of the vectord = Distance of the vector along the x-y planeα = Angle of the vector to the x-axisFrom the question,
Given:
d = 101 mα = (180+42.2) = 222.2°Substitute these values into equation 1
Y = 101sin222.2°Y = 67.84 mHence, the y component is 67.84 m.
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Water flows from the bottom of a storage tank at a rate of r(t) 200 - 4lters per minute, where OSI 50. Find the amount of water in stors that town from the tank during the first minutes Amount of water = ______ L.
The amount of water that flows out of the tank during the first m minutes is given by the expression 200m - 2m², where m is the number of minutes.
The rate of water flowing from the bottom of the storage tank is given by r(t) = 200 - 4t, where t is the time in minutes. To find the amount of water that flows out of the tank during the first m minutes, we need to integrate the rate function from t = 0 to t = m:
Amount of water = ∫₀ₘ (200 - 4t) dt
Evaluating this integral, we get:
Amount of water = [200t - 2t²] from t = 0 to t = m
Amount of water = (200m - 2m²) - (0 - 0)
Simplifying this expression, we get:
Amount of water = 200m - 2m²
Therefore, the amount of water that flows out of the tank during the first m minutes is given by the expression 200m - 2m², where m is the number of minutes.
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Given XV is 20 inches, find the length of arc XW. Leave your answer in terms of pi
The arc length XW in terms of pi is (10pi)/3.
To find the length of arc XW, we need to know the measure of the angle XDW in radians.
Since XV is the diameter of the circle, we know that angle XDV is a right angle, and angle VDW is half of angle XDW. We also know that XV is 20 inches, so its radius, XD, is half of that, or 10 inches.
Using trigonometry, we can find the measure of angle VDW:
sin(VDW) = VD/VDW
sin(VDW) = 10/20
sin(VDW) = 1/2
Since sin(30°) = 1/2, we know that angle VDW is 30 degrees (or π/6 radians). Therefore, angle XDW is twice that, or 60 degrees (or π/3 radians).
Now we can use the formula for arc length:
arc length = radius * angle in radians
So the length of arc XW is:
arc XW = 10 * (π/3)
arc XW = (10π)/3
Therefore, the arc length XW in terms of pi is (10π)/3.
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Help with problem in photo!
Check the picture below.
[tex]4+10x=\cfrac{(9x+20)+10x}{2}\implies 8+20x=19x+20\implies x=12 \\\\[-0.35em] ~\dotfill\\\\ 4+10x\implies 4+10(12)\implies \stackrel{ \measuredangle DEC }{124^o}[/tex]
Simplify this equation
Answer:
(d)
Step-by-step explanation:
Jamie mixes 2 parts of red paint with 3 parts of blue paint to make purple paint.
He uses 12 cans of blue paint.
How many cans of red paint does he use?
evaluate : 20-[5+(9-6]
Answer:
12
Step-by-step explanation:
20-(5+(9-6)) = 20-(5+(3)) = 20-(8) = 12.
Alternatively, rewrite the question without parenthesis.
20-(5+(9-6)) = 20-(5+9-6) = 20-5-9+6 = 12.
Find a set of parametric equations of the line with the given characteristics. (Enter your answers as a comma-separated list.)
The line passes through the point (-4, 8, 7) and is perpendicular to the plane given by -x + 4y + z = 8.
One possible set of parametric equations for the line is:
x = -4 + 4t
y = 8 - t
z = 7 - 4t
To see why these work, let's first consider the equation of the plane: -x + 4y + z = 8. This can also be written in vector form as:
[ -1, 4, 1 ] · [ x, y, z ] = 8
where · denotes the dot product. This equation says that the normal vector to the plane is [ -1, 4, 1 ], and that any point on the plane satisfies the equation.
Now, since the line we want is perpendicular to the plane, its direction vector must be parallel to the normal vector to the plane. In other words, the direction vector of the line must be some multiple of [ -1, 4, 1 ]. Let's call this direction vector d.
To find d, we can use the fact that the dot product of two perpendicular vectors is zero. So we have:
d · [ -1, 4, 1 ] = 0
Expanding this out, we get:
-1d1 + 4d2 + 1d3 = 0
where d1, d2, d3 are the components of d. This equation tells us that d must be of the form:
d = [ 4k, k, -k ]
where k is any non-zero scalar (i.e. any non-zero real number).
Now we just need to find a point on the line. We're given that the line passes through (-4, 8, 7), so this will be our starting point. Let's call this point P.
We can now write the parametric equations of the line in vector form as:
P + td
where t is any scalar (i.e. any real number). Substituting in the expressions for P and d that we found above, we get:
[ -4, 8, 7 ] + t[ 4k, k, -k ]
Expanding this out, we get the set of parametric equations I gave at the beginning:
x = -4 + 4tk
y = 8 + tk
z = 7 - tk
where k is any non-zero scalar.
To find a set of parametric equations for the line, we first need to determine the direction vector of the line. Since the line is perpendicular to the plane given by -x + 4y + z = 8, we can use the plane's normal vector as the direction vector for the line. The normal vector for the plane can be determined by the coefficients of x, y, and z, which are (-1, 4, 1).
Now that we have the direction vector (-1, 4, 1) and the point the line passes through (-4, 8, 7), we can write the parametric equations as follows:
x(t) = -4 - t
y(t) = 8 + 4t
z(t) = 7 + t
So, the set of parametric equations for the line is {x(t) = -4 - t, y(t) = 8 + 4t, z(t) = 7 + t}.
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As a general guideline, the research hypothesis should be stated as the:.
As a general guideline, the research hypothesis should be stated as the alternative hypothesis, which is the statement that researchers are trying to support or prove.
The research hypothesis is a statement that describes the expected relationship between variables or the expected difference between groups in a research study. It should be based on a clear and specific research question, and it should be testable using appropriate statistical methods.
In other words, the research hypothesis should be a clear and concise statement that proposes a relationship or difference between variables that can be tested through data analysis. It should also be framed in a way that allows for the rejection or acceptance of the hypothesis based on the results of the study.
The null hypothesis, on the other hand, is the statement that there is no significant relationship or difference between variables. It serves as the default assumption until evidence is provided to support the alternative hypothesis.
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A research hypothesis should be stated as the predicted outcome of the study. It's often a declarative sentence that shows the relationship between variables in a study. The research hypothesis is often contrasted with a null hypothesis, which claims no significant relationship between the study's variables.
Explanation:A Research HypothesisIn general, a research hypothesis should be stated as the predicted outcome of the study. A research hypothesis is usually written in a declarative sentence format and states the relationship between variables in the study. For example, if your research is about studying the impact of amount of study time on test scores, your hypothesis could be: 'Students who spend more time studying will have higher test scores.'
The research hypothesis is often contrasted with a null hypothesis, which states there will be no significant relationship between the study's variables. In our example, the null hypothesis would be: 'The amount of study time will not impact the test scores significantly.' Remember, a research hypothesis should always be testable through research methods.
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A new plane can travel 1200000 m in 120 minutes. Find its speed in km/h.
Answer:
Step-by-step explanation:
We can start by converting the distance and time to the appropriate units.
1200000 meters = 1200 kilometers (since 1 kilometer = 1000 meters)
120 minutes = 2 hours (since 1 hour = 60 minutes)
Now we can use the formula:
speed = distance / time
speed = 1200 km / 2 hours
speed = 600 km/h
Therefore, the speed of the new plane is 600 km/h.
Answer: 600km/
First step:
1200000m=1200Km * 1m=0,001km
Second step:
120min=2h *1h=60min
Last step:
1200km÷2h= 600km/
SOLUTION
600km/
Step-by-step explanation:
Aria drank 500 milliliters of water after her run. her best friend, andrea, drank 0.75 liter of water. who drank more?group of answer choices
Determine whether the graph shows a positive correlation, a negative correlation, or no correlation. If there is a positive or negative correlation, describe its meaning in the situation. Domestic Traveler Spending in the U.S., 1987-1999 Spending (dollars in billions) A graph titled Domestic Traveler Spending in the U S from 1987 to 1999 has year on the x-axis, and spending (dollars in billions) on the y-axis, from 225 to 450 in increments of 25. Year Source: The World Almanac, 2003 a. positive correlation; as time passes, spending increases. b. no correlation c. positive correlation; as time passes, spending decreases. d. negative correlation; as time passes, spending decreases.
There is a positive correlation and as such as time passes, spending increases.
Checking the correlation of the graphThe descriptions of the graph from the question are given as
Year (x - axis): 1987 to 1999Spending (y - axis, dollars in billions) 225 to 450 in increments of 25.From the above statements, we can make the following summary
As the year increase, the spending also increase
The above summary is about the correlation of the graph
And it means that there is a positive correlation and as such as time passes, spending increases.
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Solve the problem.
Find the area bounded by y = 3 / (√36-9x^2) • X = 0, y = 0, and x = 3. Give your answer in exact form.
To solve the problem, we first need to graph the equation y = 3 / (√36-9x^2) and find the points where it intersects the x-axis and y-axis.
To find the x-intercept, we set y = 0 and solve for x:
0 = 3 / (√36-9x^2)
0 = 3
This has no solution, which means that the graph does not intersect the x-axis.
To find the y-intercept, we set x = 0 and solve for y:
y = 3 / (√36-9(0)^2)
y = 3 / 6
y = 1/2
So the graph intersects the y-axis at (0, 1/2).
Next, we need to find the point where the graph intersects the vertical line x = 3. To do this, we substitute x = 3 into the equation y = 3 / (√36-9x^2):
y = 3 / (√36-9(3)^2)
y = 3 / (√-243)
This is undefined, which means that the graph does not intersect the line x = 3.
Now we can draw a rough sketch of the graph and the region bounded by the x-axis, the line x = 0, and the curve y = 3 / (√36-9x^2):
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The area we want to find is the shaded region, which is bounded by the x-axis, the line x = 0, and the curve y = 3 / (√36-9x^2). To find the area, we need to integrate the equation y = 3 / (√36-9x^2) with respect to x from x = 0 to x = 3:
A = ∫(0 to 3) 3 / (√36-9x^2) dx
We can simplify this integral by using the substitution u = 3x, du/dx = 3, dx = du/3:
A = ∫(0 to 9) 1 / (u^2 - 36) du/3
Next, we use partial fractions to break up the integrand into simpler terms:
1 / (u^2 - 36) = 1 / (6(u - 3)) - 1 / (6(u + 3))
So we have:
A = ∫(0 to 9) (1 / (6(u - 3))) - (1 / (6(u + 3))) du/3
A = (1/6) [ln|u - 3| - ln|u + 3|] from 0 to 9
A = (1/6) [ln(6) - ln(12) - ln(6) + ln(6)]
A = (1/6) [ln(1/2)]
A = (-1/6) ln(2)
Therefore, the exact area bounded by y = 3 / (√36-9x^2), x = 0, y = 0, and x = 3 is (-1/6) ln(2).
To find the area bounded by y = 3 / (√36-9x^2), x = 0, y = 0, and x = 3, we can set up an integral to compute the definite integral of the function over the given interval [0, 3]. The integral will represent the area under the curve:
Area = ∫[0, 3] (3 / (√(36-9x^2))) dx
To solve the integral, perform a substitution:
Let u = 36 - 9x^2
Then, du = -18x dx
Now, we can rewrite the integral:
Area = ∫[-√36, 0] (-1/6) (3/u) du
Solve the integral:
Area = -1/2 [ln|u|] evaluated from -√36 to 0
Area = -1/2 [ln|0| - ln|-√36|]
Area = -1/2 [ln|-√36|]
Since the natural logarithm of a negative number is undefined, there's an error in the original problem. Check the problem's constraints and the given function to ensure accuracy before proceeding.
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