To evaluate the integral of (3x+6)/(2x²-3)dx, we can use partial fraction decomposition:
(3x+6)/(2x²-3) = A/(x-√(3)/2) + B/(x+√(3)/2)
Multiplying both sides by the denominator and simplifying, we get:
3x+6 = A(x+√(3)/2) + B(x-√(3)/2)
Setting x = √(3)/2, we get:
3√(3)/2 + 6 = B(√(3)/2-√(3)/2) = 0
So B = -2√(3). Setting x = -√(3)/2, we get:
-3√(3)/2 + 6 = A(-√(3)/2+√(3)/2) = 0
So A = 2√(3). Therefore, we have:
(3x+6)/(2x^2-3) = 2√(3)/(x-√(3)/2) - 2√(3)/(x+√(3)/2)
Integrating each term, we get:
∫(3x+6)/(2x²-3)dx = 2√(3)ln|x-√(3)/2| - 2√(3)ln|x+√(3)/2| + C
where C is the arbitrary constant.
To evaluate the integral of the function 3x + 6 with respect to x, we will use the integral symbol and find the antiderivative:
∫(3x + 6) dx
To find the antiderivative, we will apply the power rule, which states that the integral of x^n is (x^(n+1))/(n+1), and the constant rule, which states that the integral of a constant is the constant times the variable:
(3 * (x^(1+1))/(1+1)) + (6 * x) + C
Simplifying the expression:
(3x²)/2 + 6x + C
Here, C is the arbitrary constant. So, the evaluated integral of 3x + 6 is:
(3x²)/2 + 6x + C
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lmno is a parallelogram. if nm = x + 27 and ol = 3x + 9 find the value of x and then find nm and ol.
The length of NM and OL in the parallelogram LMNO are both 36 units.
How we find the value of x?Since LMNO is a parallelogram, we know that opposite sides are equal in length. Therefore, we can set NM and OL equal to each other:
NM = OL
We also know that NM = x + 27 and OL = 3x + 9, so we can substitute these expressions into the equation:
x + 27 = 3x + 9
Solving for x:
2x = 18
x = 9
Now that we know x, we can substitute it back into the expressions for NM and OL:
NM = x + 27 = 9 + 27 = 36
OL = 3x + 9 = 3(9) + 9 = 36
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3.
Noah is playing a game where he must spin two wheels, each with 9 equal slices. There are 3 red slices, 3 green slices, 2 blue slices and 1 yellow slice on each wheel. If Noah spins and lands on a yellow slice on both wheels he wins, but if he lands on any other color, he loses. This information was used to create the following area model.
Is this a fair game? Why or why not?
No, the game is not fair because Noah does not have equal probabilities of winning or losing.
No, the game is not fair because Noah has equal probabilities of winning or losing.
Yes, the game is fair because Noah has equal probabilities of winning or losing.
Yes, the game is fair because Noah does not have equal probabilities of winning or losing
The answer to whether it is this a fair game is: No, the game is not fair because Noah does not have equal probabilities of winning or losing. Therefore, the correct option is 1.
The reason why it is not a fair game is as follows.
There are 9 slices on each wheel, so the total possible outcomes when spinning both wheels are 9 x 9 = 81.To win, Noah needs to land on a yellow slice on both wheels. There's only 1 yellow slice on each wheel, so the probability of this happening is 1/9 (for the first wheel) multiplied by 1/9 (for the second wheel), which is 1/81.The probability of losing is the opposite, meaning he doesn't land on a yellow slice on either wheel. The probability of not landing on a yellow slice on one wheel is 8/9. So, the probability of losing is 8/9 (for the first wheel) multiplied by 8/9 (for the second wheel), which is 64/81.Since the probabilities of winning and losing are not equal (1/81 vs 64/81), the game is not fair. Therefore, the correct answer is option 1.
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HELP PLEASE BRAINLIEST + POINTS
Answer:
CD = 34 units--------------------------
Since CD is diameter, therefore the angle CAD opposite to it is a right angle.
We are given the lengths of two legs, AD = 16 and AC = 30.
Use Pythagorean theorem to find the length of the hypotenuse CD:
CD² = AD² + AC²CD² = 16² + 30²CD² = 1156CD = √1156CD = 34Verónica jogged 10 3/16 miles in a one week, the next week she jogged 8 7/16 miles. how many more miles did she jog the first week? pls answer
Verónica jogged 7/4 or 1 and 3/4 more miles in the first week than in the second week.
Verónica jogged 10 3/16 miles in one week, next week she jogged 8 7/16 miles. how many miles did she jog the first week?Verónica jogged 10 3/16 miles in the first week and 8 7/16 miles in the second week. To find how many more miles she jogged in the first week, we need to subtract the distance she jogged in the second week from the distance she jogged in the first week:
10 3/16 miles - 8 7/16 miles
We need to first convert both mixed numbers to improper fractions:
10 3/16 = (10 x 16 + 3) / 16 = 163 / 16
8 7/16 = (8 x 16 + 7) / 16 = 135 / 16
Now we can subtract the two fractions:
163 / 16 - 135 / 16 = (163 - 135) / 16 = 28 / 16
We can simplify the fraction by dividing both the numerator and the denominator by their greatest common factor (GCF), which is 4:
28 / 16 = (4 x 7) / (4 x 4) = 7 / 4
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What is the surface area of the cylinder? Approximate using π = 3.14 and round to the nearest square meter.
a cylinder has radius labeled 2.2 meters and height labeled 6.5 meters
99 square meters
105 square meters
117 square meters
120 square meters
The surface area of the cylinder, rounded to the nearest square meter, is: D. 120 square meters
What is the Surface Area of a Cylinder?The surface area of a cylinder can be calculated by using the formula expressed as:
SA = 2πr(h + r), where r is the radius and h is the height.
Given the following:
π = 3.14
radius (r) = 2.2 m
height of the cylinder (h) = 6.5 m
Plug in the values:
SA = 2 * 3.14 * 2.2 * (6.5 + 2.2)
SA ≈ 120 square meters (rounded to the nearest square meter)
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The cone and the sphere shown have the same volume. The diameter of the cone is 24 cm, and the diameter of the sphere is 18 cm. What is the height h of the cone?
40.50 cm
2.25 cm
6.75 cm
20.25 cm
Answer:
i think the answer is 20.25
Step-by-step explanation:
The monthly income of a man is Rs 53000. He deposits 20% of his yearly income in civil investment fund and 10% in charity. If 1 % social security tax should be paid on First Rs 300000and 15% tax is imposed yearly ,how much tax should he pay?
The man needs to pay a total tax of Rs 69780 (3000 for social security and 66780 for yearly tax).
How to find the tax should he pay?To find the tax, let's find the man's yearly income:
Yearly income = Monthly income x 12
Yearly income = 53000 x 12 = 636000
Next, let's find how much he deposits in civil investment fund and charity:
Amount deposited in civil investment fund = Yearly income x 20%
Amount deposited in civil investment fund = 636000 x 0.2 = 127200
Amount deposited in charity = Yearly income x 10%
Amount deposited in charity = 636000 x 0.1 = 63600
Now, let's calculate the total taxable income:
Total taxable income = Yearly income - Amount deposited in civil investment fund - Amount deposited in charity
Total taxable income = 636000 - 127200 - 63600 = 445200
Since the man's taxable income is above Rs 300000, he needs to pay 1% social security tax on Rs 300000:
Social security tax = 1% of 300000 = 3000
Now, let's calculate the yearly tax imposed at a rate of 15%:
Yearly tax = Total taxable income x 15%
Yearly tax = 445200 x 0.15 = 66780
Therefore, the man needs to pay a total tax of Rs 69780 (3000 for social security and 66780 for yearly tax).
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The altitude (or height) of a triangle is increasing at a rate of 2.5cm/min while the area of the triangle is increasing at a rate of 3cm2/min. At what rate (in cm/min) is the base of the triangle changing when the altitude is 12cm and the area is 84cm2 Round your answer to three decimal places.
The base is decreasing at 1.917 cm/min when altitude is 12cm and area=84cm².
Let A be the area of the triangle, h be the height of the triangle, and b be the base of the triangle.
Then the formula for the area of a triangle is:
A = (1/2)bh
We are given that dh/dt = 2.5 cm/min (the height is increasing at a rate of 2.5cm/min), and dA/dt = 3 cm²/min (the area is increasing at a rate of 3cm²/min).
We want to find db/dt, the rate of change of the base of the triangle when h = 12 cm and A = 84 cm².
To solve this problem, we need to use the chain rule of differentiation.
We start by differentiating both sides of the formula for the area of a triangle with respect to time t:
dA/dt = (1/2) d/dt (bh)
Next, we can use the product rule of differentiation to find d/dt (bh):
d/dt (bh) = b dh/dt + h db/dt
Substituting this into the previous equation gives:
dA/dt = (1/2) [ b dh/dt + h db/dt ]
Now we can substitute the given values of dh/dt and dA/dt, as well as h = 12 cm and A = 84 cm².
To find db/dt:
3 cm²/min = (1/2) [ b (2.5 cm/min) + 12 cm db/dt ]
Simplifying this expression gives:
6 cm²/min = 2.5 b cm²/min + 12 cm db/dt
Substituting A = 84 cm² and h = 12 cm into the formula for the area of a triangle gives:
84 cm² = (1/2) b (12 cm)
Simplifying this expression gives:
b = 14 cm
Now we can substitute b = 14 cm into the previous equation to find db/dt:
6 cm²/min = 2.5 (14 cm) cm²/min + 12 cm db/dt
Simplifying this expression gives:
db/dt = (6 cm²/min - 35 cm²/min) / (12 cm)
db/dt = -1.917 cm/min (rounded to three decimal places)
Therefore, the base of the triangle is decreasing at a rate of 1.917 cm/min when the height is 12 cm and the area is 84 cm².
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Find the value of x
Hellpppp
Answer:
r = 14.60
Step-by-step explanation:
r²= 9²+(23/2)²
= 9²+11.5²
= 81+132.25
r² = 213.25
r = √213.25
= 14.60 to 2d.p
A hand-made carton has the following dimensions:
length of the base-7 inches
width of the base-6 inches
height of the carton-6 inches
what change should be made to the dimensions to increase the
volume of the carton by 42 cubic inches?
a. increase the height of the carton to 7 inches
b. increase the height of the carton to 8 inches
c. increase the width of the base to 8 inches
d. increase the width of the base to 9 inches
The correct answer is option (a), increase the height of the carton to 7 inches.
How can the volume of a handmade carton be increased by 42 cubic inches?
To increase the volume of the carton by 42 cubic inches, we need to increase either the length, width, or height of the carton or a combination of these dimensions.
Let's first calculate the current volume of the carton:
Volume = length x width x height
Volume = 7 x 6 x 6
Volume = 252 cubic inches
Now, we need to find a new dimension that will increase the volume by 42 cubic inches.
a) If we increase the height to 7 inches, the new volume will be:
New Volume = 7 x 6 x 7
New Volume = 294 cubic inches
The volume has increased by 42 cubic inches, so option (a) is the correct answer.
b) If we increase the height to 8 inches, the new volume will be:
New Volume = 7 x 6 x 8
New Volume = 336 cubic inches
The volume has increased by 84 cubic inches, which is more than required.
c) If we increase the width of the base to 8 inches, the new volume will be:
New Volume = 7 x 8 x 6
New Volume = 336 cubic inches
The volume has increased by 84 cubic inches, which is more than required.
d) If we increase the width of the base to 9 inches, the new volume will be:
New Volume = 7 x 9 x 6
New Volume = 378 cubic inches
The volume has increased by 126 cubic inches, which is more than required.
Therefore, the correct answer is option (a), increase the height of the carton to 7 inches.
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Carson decides to estimate the volume of a coffee cup by modeling it as a right cylinder. Carson measures its circumference as 15.1 cm and its volume as 161 cubic centimeters. Find the height of the cup in centimeters. Round your answer to the nearest tenth if necessary.
please help ;-;
To find the height of the coffee cup, we can use the formula for the volume of a cylinder:
V = πr^2h
where V is the volume, r is the radius, and h is the height.
We are given that the circumference of the coffee cup is 15.1 cm. The formula for the circumference of a cylinder is:
C = 2πr
where C is the circumference and r is the radius.
We can use this formula to find the radius of the coffee cup:
15.1 cm = 2πr
r = 15.1 cm / (2π)
r ≈ 2.4 cm
Now we can use the given volume and radius to find the height of the coffee cup:
161 cm^3 = π(2.4 cm)^2h
h = 161 cm^3 / (π(2.4 cm)^2)
h ≈ 4.0 cm
Therefore, the height of the coffee cup is approximately 4.0 cm.
Can you help me with part C? Please
Answer: -40
Step-by-step explanation:
Rate of change is calculated as the slope. The formula for Slope is:
S = [tex]\frac{x_{1}-x_{2} }{ y_{1}-y_{2}}[/tex]
The two points we have are when x = 3 and 6.
the points are (3, 120) and (6, 0), as we can see.
plugging into the slope formula:
S = [tex]\frac{120-0 }{ 3-6}[/tex]
S = 120/-3
S = -40
Which hopefully makes sense, because the slope is negative, (the graph is falling).
Si al triple de la edad que tengo, se quita mi edad aumentada en 8 años, tendría 36 años. ¿Qué edad tengo?
damePor lo tanto, la edad que tienes es de aproximante 14.67 años.
Si al triple de la edad que tengo, se quita mi edad aumentada en 8 años, tendría 36 años. ¿
Podemos plantear este problema como una ecuación algebraica. Si llamamos "x" a la edad que tienes, la ecuación sería:
3x - 8 = 36
Ahora, despejamos la variable "x" para encontrar su valor:
3x = 36 + 8
3x = 44
x = 44/3
.Este resultado nos indica que nuestra edad actual es de aproximadamente 14.67 años. Es importante tener en cuenta que la solución no es un número entero, lo cual puede parecer inusual para una edad, pero es una respuesta matemáticamente correcta según la ecuación planteada en el problema.
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A rocket is launched from a tower. The height of the rocket, y in feet, is related to the time after launch, x in seconds, by the given equation. Using this equation, find the maximum height reached by the rocket, to the nearest tenth of a foot. Y
=
−
16
x
2
+
180
x
+
63
y=−16x 2
+180x+63
The maximum height reached by the rocket is approximately 504.6 feet.
To find the maximum height reached by the rocket, we need to determine the vertex of the parabola represented by the given quadratic equation: y = -16x^2 + 180x + 63.
The x-coordinate of the vertex can be found using the formula x = -b / 2a, where a = -16 and b = 180.
x = -180 / (2 * -16) = 180 / 32 = 5.625
Now, we'll plug the x-coordinate back into the equation to find the y-coordinate (maximum height).
y = -16(5.625)^2 + 180(5.625) + 63
y ≈ 504.6
The maximum height reached by the rocket is approximately 504.6 feet.
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A cat darts around a room chasing a ball. The cat first travels along the vector −1, 2 and then chases the ball along the vector 2, 6 − . The cat darts after the ball 1.5 times along the vector 4, 3 . This is where the cat catches the ball and chews on it. What vector describes the cat’s final position? Show all your work.
To find the cat's final position, we need to add up all the vectors representing the cat's movements.
The cat first travels along the vector −1, 2.
Next, the cat chases the ball along the vector 2, 6 − , which we can write as (2, 6) − (0, 1) = (2, 5).
Then, the cat darts after the ball 1.5 times along the vector 4, 3, which we can write as 1.5(4, 3) = (6, 4.5).
Finally, the cat's position after catching the ball is the sum of all these vectors:
(-1, 2) + (2, 5) + (6, 4.5) = (7, 11.5)
Therefore, the vector describing the cat's final position is (7, 11.5).
A social scientist is interested in determining if there is a significant difference in the proportion of republicans between two areas of town. He takes independent random samples of 200 families in each area of town and a significance test was conducted. The p-value was 0. 416. What should be our conclusions?.
The p-value of 0.416 indicates that there is no significant difference in the proportion of Republicans between the two areas of town. Therefore, we fail to reject the null hypothesis and conclude that there is no evidence of a significant difference in the proportion of Republicans between the two areas of town.
Based on the given information, the p-value is 0.416, which is larger than the conventional level of significance (e.g., 0.05 or 0.01). Therefore, we fail to reject the null hypothesis that there is no significant difference in the proportion of republicans between the two areas of town.
In other words, we cannot conclude that there is a significant difference between the two areas. It is possible that any observed difference could be due to chance.
However, it is important to note that statistical significance does not necessarily mean practical significance, and further investigation may be needed to determine if there are any meaningful differences between the two areas in terms of the proportion of republicans.
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To the nearest whole cubic centimeter, what is the volume of the prism?
The volume of the prism in 120 cubic centimeter.
What is volume?Volume is a measure of the amount of space occupied by a three-dimensional object. It is a scalar quantity that characterizes the "size" or "capacity" of an object in three dimensions.
What is prism?A prism is a three-dimensional geometric shape that has two parallel and congruent bases connected by lateral faces that are typically rectangular or parallelogram-shaped. Prisms are classified based on the shape of their base, such as rectangular prisms, triangular prisms, pentagonal prisms, hexagonal prisms, and so on. The lateral faces of a prism are perpendicular to the bases, and the bases are usually parallel to each other.
According to the given information:
Given the measures of prism is length = 8cm, breadth = 6 cm, height = 5 cm
The formula for volume of prism = 1/2 l*b*h
= 1/2 8*6*5
= 120 cubic centimeter
Hence the volume of prism is 120 cubic centimeter.
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What is the product? assume x greater-than-or-equal-to 0 (startroot 3 x endroot startroot 5 endroot) (startroot 15 x endroot 2 startroot 30 endroot) 3 x startroot 5 endroot 3 startroot 165 x endroot 10 startroot 6 endroot 3 x startroot 5 endroot 6 startroot 10 x endroot 5 startroot 3 x endroot 10 startroot 6 endroot 3 x startroot 5 endroot 10 startroot 6 endroot startroot 3 x endroot 5 startroot 3 x endroot 10 startroot 6 endroot
The product of the given expression is 2,916,000,000x³√(9,900x²).
The given expression contains several terms with roots and variables. To simplify and find the product, we'll first multiply the terms with similar roots and variables. The expression is:
√(3x)√5 √(15x)√2 √(30) 3x√5 3√(165x) √10 √6 3x√5 √6 √(10x) √5 √(3x) √10 √6 3x√5 √10 √6 √(3x) √5 √(3x) √10 √6
We can group terms with the same roots and variables together:
(√(3x))⁴ (3x)³ (√5)⁴ (√10)³ (√6)³ √15x √2 √30 √165x
Now, we can simplify each group:
81x³ * 625 * 1000 * 216 * √(2 * 15x * 30 * 165x)
Combine the constants and variables under the root:
81x³ * 625 * 1000 * 216 * √(9,900x²)
Calculate the product of the constants:
13,500,000 * 216 = 2,916,000,000
So, the final simplified expression is:
2,916,000,000x³√(9,900x²)
In summary, the product of the given expression is 2,916,000,000x³√(9,900x²).
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Trucks are delivering gravel to a construction site.
Each truck holds 7.5 cubic yards of gravel.
The weight of one cubic yard of gravel is 1.48 tons
The gravel will be placed in containers that each holds 3.7 tons of gravel.
How many containers of this size are needed to hold all the gravel from one truck.
Please some one answer this with work shown, i need to show work!! Thank you
To determine how many containers of size 3.7 tons are needed to hold all the gravel from one truck, we need to first calculate how many tons of gravel are in one truck.
How many containers of this size are needed to hold all the gravel from one truck?Since each truck holds 7.5 cubic yards of gravel, and the weight of one cubic yard of gravel is 1.48 tons, we can calculate the total weight of gravel in one truck as follows:
7.5 cubic yards x 1.48 tons per cubic yard = 11.1 tons
Therefore, each truck carries 11.1 tons of gravel.
To determine how many containers of size 3.7 tons are needed to hold all the gravel from one truck, we can divide the total weight of gravel in one truck by the capacity of each container:
11.1 tons ÷ 3.7 tons per container = 3 containers
Therefore, three containers of size 3.7 tons are needed to hold all the gravel from one truck.
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Which cardboard box can hold the greatest number of 1 in x 2 in x 4 in sponges
The cardboard box with the largest volume can hold the greatest number of 1 in x 2 in x 4 in sponges.
To find the box with the largest volume, first determine the volume of each sponge: V_sponge = 1 in x 2 in x 4 in = 8 cubic inches. Next, find the volume of each box by multiplying its length, width, and height (V_box = L x W x H).
To determine how many sponges each box can hold, divide the volume of the box by the volume of the sponge (V_box / V_sponge). The box with the highest resulting quotient can hold the most 1 in x 2 in x 4 in sponges.
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The area of a circle increases at a rate of 2 cm2/s. a. How fast is the radius changing when the radius is 4 cm? b. How fast is the radius changing when the circumference is 3 cm?
a) When the radius is 4 cm, it is changing at a rate of 1/(4π) cm/s.
b) When the circumference is 3 cm, the radius is changing at a rate of 2/3 cm/s.
How to find the change of radiusa. Given that the area of a circle increases at a rate of 2 cm²/s, let's denote this rate as dA/dt.
The formula for the area of a circle is A = πr²,
where A is the area and r is the radius.
We want to find the rate at which the radius is changing, or dr/dt, when the radius is 4 cm.
Using implicit differentiation with respect to time t, we get:
dA/dt = d(πr²)/dt 2 = 2πr(dr/dt)
Now, we'll plug in the radius value of 4 cm:
2 = 2π(4)(dr/dt)
Solving for dr/dt, we get:
dr/dt = 1/(4π) cm/s
b. We are given the circumference, which is 3 cm.
The formula for the circumference of a circle is C = 2πr, where C is the circumference and r is the radius.
First, we need to find the radius when the circumference is 3 cm: 3 = 2πr r = 3/(2π)
Now, we'll plug this value for the radius back into the formula from part a:
2 = 2π(3/(2π))(dr/dt)
Solving for dr/dt, we get:
dr/dt = 2/3 cm/s
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Does anyone know the answer
Answer:
B) <FBG
Step-by-step explanation:
An adjacent angle is an angle that is right next to the given angle.
In this case, the given angle is <EBF.
We can see that the only 2 options here for an adjacent set of angles is either <FBG or <EBD.
Looking at the options, we can only see that <FBG is an option, making B the correct option.
Hope this helps :)
An author works on a new book and after writing a few chapters, begins a new plan to write 600
words per day. On the fifth day of working on this plan. 4,500 words have been written.
If the equation y=600. 0 + b represents the total number of words the author has written, y, based
on the number of days, x, since the new plan was started, what is the value of b?
The value of b is 1,500, which means that the author had already written 1,500 words before starting the new plan.
In the equation y = 600x + b, x represents the number of days since the author started the new plan to write 600 words per day, and y represents the total number of words written.
After the fifth day, the author has written 4,500 words. Thus, we can substitute x = 5 and y = 4,500 into the equation and solve for b:
4,500 = 600(5) + b
4,500 = 3,000 + b
b = 4,500 - 3,000
b = 1,500
Therefore, the value of b is 1,500, which means that the author had already written 1,500 words before starting the new plan.
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ASAP plis i need an answer and explanation
Answer: Z=21, Y=9, X=1
Step-by-step explanation:
Z---------
Since 3z and 63 are equal angles, then 3z must equal 63 as well. So 3 times what is 63? 3 times 21. Z=21.
X------1
2 times 1 is 2.
3 times 1 is 3. 3-1 is 2.
x=1
Y----
Same concept as Z. Since the length 13 and Y+4 are equal sides (definition of the kite figure I forget the proper name) then 13-4 is your answer. 9.
Answer the following:
Explain how you know that y directly relates to x in the given table. Determine the constant of variation, k.
Write an equation for the direct variation
The equation for the direct variation is y = 2x. This the equation that directly relates y to x. The value of k is 2.
To know that y directly relates to x in a table, we need to check if y increases or decreases proportionally with x. In the given table, we can see that as x increases, y also increases. This indicates a direct relationship between x and y.
The constant of variation, k, can be determined by dividing any y value by its corresponding x value. Let's choose the first row of the table: y=4, x=2. Therefore, k = y/x = 4/2 = 2.
Now, we can write an equation for the direct variation: y = kx. Plugging in the value of k, we get y = 2x. This equation shows that y is directly proportional to x, with a constant of variation, k, equal to 2.
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Ailani draws a map of her local town. she places the town hall at the origin of a coordinate plane and represents a lake with a circle drawn on the map. the center of the lake is 19 miles east and 3 miles south of the town hall, and the radius of the lake is 0. 5 miles. if the positive x-axis represents east and the positive y-axis represents north, which equation represents the lake? (x 19)2 (y – 3)2 = 0. 5 (x – 19)2 (y 3)2 = 0. 5 (x 19)2 (y – 3)2 = 0. 25 (x – 19)2 (y 3)2 = 0. 25.
The equation is (x^2 + y^2 - 38x + 6y = -369).
The center of the lake is 19 miles east and 3 miles south of the town hall, which means the coordinates of the center are (19,-3). The radius of the lake is 0.5 miles.
Using the standard equation of a circle, we have:
(x - h)^2 + (y - k)^2 = r^2
where (h,k) is the center of the circle and r is the radius.
Substituting the given values, we get: (x - 19)^2 + (y + 3)^2 = 0.5^2
Expanding the left side, we get: x^2 - 38x + 361 + y^2 + 6y + 9 = 0.25
Simplifying and rearranging terms, we get:
x^2 + y^2 - 38x + 6y + 369.25 = 0.25
Subtracting 369 from both sides, we get:
x^2 + y^2 - 38x + 6y = -369
Therefore, the equation that represents the lake on the map is:
(x - 19)^2 + (y + 3)^2 = 0.5^2, which can be simplified to (x^2 + y^2 - 38x + 6y = -369).
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Find the measure of ZB 75° b
Answer: ∠b = 105°
Step-by-step explanation:
We know that a straight line is equal to 180 degrees. We will create an equation and solve for ∠b.
180° = ∠b + 75°
∠b = 180° - 75°
∠b = 105°
Find < F:
(Round your answer to the nearest hundredth)
The length of the hypotenuse is approximately 7.21 ft.
To solve this problem, we can use the Pythagorean theorem, which states that in a right triangle, the sum of the squares of the two shorter sides (legs) is equal to the square of the longest side (hypotenuse). In mathematical terms, it looks like this:
a² + b² = c²
Where "a" and "b" are the lengths of the legs, and "c" is the length of the hypotenuse.
In your case, we can substitute the given values into the equation:
6² + 4² = c²
Simplifying:
36 + 16 = c²
52 = c²
To solve for "c," we need to take the square root of both sides of the equation:
√(52) = c
We can simplify the square root of 52 to be 2 times the square root of 13. Therefore:
c ≈ 7.21 ft
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Complete Question:
Find the value of hypotenuse of the given triangle by using the Pythagoras theorem.
The angle of elevation between a fishing vessel and the top of a 50-meter-tall lighthouse is 12 degrees. What is the approximate distance between the fishing vessel and the base of the lighthouse?
A.
10. 6 meters
B.
48. 9 meters
C.
235. 2 meters
D.
240. 5 meters
We solve this problem using the angle of elevation, we can apply the tangent function from trigonometry. The approximate distance between the fishing vessel and the base of the lighthouse is 235.2 meters, which corresponds to option C. 235. 2 meters
Find the approximate distance between the fishing vessel and the base of the 50-meter-tall lighthouse when the angle of elevation is 12 degrees.
Set up the equation using tangent function.
tan(angle of elevation) = (height of lighthouse) / (distance between vessel and lighthouse base)
Plug in the values.
tan(12°) = 50 / distance
Solve for the distance.
distance = 50 / tan(12°)
Calculate the distance using a calculator.
distance ≈ 235.2 meters
So, the approximate distance between the fishing vessel and the base of the lighthouse is 235.2 meters, which corresponds to option C. 235. 2 meters
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express as a single simplified fraction. 3m^2-3n^2/m^2+mp divided by 6m-6n/p+m
The single simplified fraction is (m + n)(p + m) / 2m.
To simplify the expression
[tex](3m^2 - 3n^2) / (m^2 + mp)÷ (6m - 6n) / (p + m)[/tex]
we need to invert the second fraction and multiply by the first.
[tex](3m^2 - 3n^2) / (m^2 + mp) \times (p + m) / (6m - 6n)[/tex]
We can then factor out a 3 from the numerator and the denominator, and cancel out the (m - n) terms. 3(m + n)(m - n) / 3m(m - n) x (p + m) / 6(m - n)
Simplifying further, we can cancel out the 3's and the (m - n) terms. (m + n) / m x (p + m) / 2
The simplified expression is (m + n)(p + m) / 2m.
To simplify the given expression, we invert the second fraction and multiply it by the first. Then we factor out common terms and cancel out like terms. We simplify the expression to obtain the single fraction (m + n)(p + m) / 2m.
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