Let u = tan⁻¹x, then du/dx = 1/(1+x²).
Using the formula for the derivative of inverse tangent, we have:
tan(u) = x
sec²(u) du/dx = 1
du/dx = cos²(u)
Substituting into the original expression, we get:
∫(tan⁻¹x × e^tan⁻¹x)/(1+x²) dx = ∫(tan⁻¹x × e^u × cos²(u)) du
Using integration by parts with u = tan⁻¹x and dv = e^u × cos²(u) du, we get:
v = (1/2) e^u (sin(u) + u cos(u))
∫(tan⁻¹x × e^u × cos²(u)) du = (1/2) e^u (sin(u) + u cos(u)) tan⁻¹x - ∫[(sin(u) + u cos(u)) / (1+x²)] dx
Substituting back u = tan⁻¹x, we get:
∫(tan⁻¹x × e^tan⁻¹x × cos²(tan⁻¹x)) dx = (1/2) e^tan⁻¹x (x sin(tan⁻¹x) + cos(tan⁻¹x)) tan⁻¹x - ∫[(x cos(tan⁻¹x) + sin(tan⁻¹x)) / (1+x²)] dx
Using the identity sin(tan⁻¹x) = x / √(1+x²) and cos(tan⁻¹x) = 1 / √(1+x²), we simplify the expression to:
∫(tan⁻¹x × e^tan⁻¹x × cos²(tan⁻¹x)) dx = (1/2) x e^tan⁻¹x + (1/2) ∫[e^tan⁻¹x / (1+x²)] dx
The remaining integral can be solved using another substitution with v = tan⁻¹x, which results in:
∫(tan⁻¹x × e^tan⁻¹x × cos²(tan⁻¹x)) dx = (1/2) x e^tan⁻¹x + (1/2) ln(1+x²) + C, where C is the constant of integration
Which coordinate is on the graph of the equation 4x - 3y = 8?
O A. (1,1)
O B. (4.2)
O C. (2.-5)
OD.
(6,5)
Answer:C
Step-by-step explanation:
Please help me......................
Answer:
Your answer will be y=4 I believe
Please mark me brainliest if that is correct :)
2. Find the surface area of the triangular prism whose net is shown below.
Show all work.
12 cm
13 cm
5 cm
12 cm
8 cm
8 cm
13 cm
5 cm
12 cm
12 cm
Answer:
133
Step-by-step explanation:
Angie wants to earn a 90% for her overall history grade. During the course
she has to take 5 tests. So far, she has only taken 4 tests. Her exam
scores for the tests are as follows: 85, 83, 99, 95. If Angie wants to ensure
she will receive a 90%, what is the minimum score on the 5th test that she
can receive?
Brandon invested $9,200 in an account paying an interest rate of 3 1/4%
compunded quarterly. Lamonte invested $9,200 in an account paying an interest rate of 2 7/8% compounded monthly. After 19 years, how much more money would Brandon have in his account than Lamonte, to the nearest dollar?
Answer:
$1141
Step-by-step explanation:
The general formula for compound interest is:
[tex]A(t)=P(1+r/n)^n^t[/tex], where A is the amount in dollars, P is the principal (i.e., amount invested), r is the interest rate, n is the number of compounding periods, and t is the time in years.
For Brandon's account (and equation), P = $9200, r = 0.0325, n = 4 and t = 19.
Explanations for r and n:
We always convert the percentage to a decimal when dealing with interest. 3 1/4% is the same as 3.25%. We find the decimal by moving the decimal point two places to the right or by dividing 3.25 by 100 which gives us r = 0.0325Quarterly divides a unit into 4. There are 12 months in a year and dividing 12 into 4 means that the money is compounded every 3 monthsNow, we can simply plug in everything to find the amount Brandon would have in 19 years and round to the nearest dollar (i.e., nearest whole number):
[tex]A_{Brandon}(19)=9200(1+0.0325/4)^(^4^*^1^9^)\\ A_{Brandon}(19)=9200(1.008125)^7^6\\ A_{Brandon(19)}=17016.92431=17017[/tex]
For Lamonte's account (and equation), P = $9200, r = 0.02875, n = 12, and t = 19
Explanations for r and n:
2 7/8% is the same as 2.875% and when we move the decimal two places to the left (or divide 2.875 by 100), our decimal for r is 0.02875There are 12 months in year so monthly means that the money is compounded every month. This is why n = 12 when money is compounded monthlyWe can again plug in the values into the general equation to find the amount Lamonte would have in 19 years and round to the nearest dollar/whole number:
[tex]A_{Lamonte}(19)=9200(1+0.02875/12)^(^1^2^*^1^9^)\\ A_{Lamonte}(19)=9200(1.002395833)^2^2^8\\A_{Lamonte}(19)=15875.86592=15876[/tex]
Now we can subtract Lamonte's amount (15875) from Brandon's (17017) to find how much more Brandon would have in his account than Lamonte in 19 years:
17017 - 15876 = $1141
**If you didn't do any intermediate rounding and waited until the very last subtraction to round, you'd still get a similar answer
17016.92431 - 15875.86713 = 1141.05718 = $1141
Describe the correlation for an r-value of –0.86.
A.
weak, negative correlation
B.
strong, negative correlation
C.
strong, positive correlation
D.
weak, positive correlation
B. strong, negative correlation. An r-value of -0.86 shows a significant, negative correlation between the two variables under consideration.
An r-value of -0.86 shows a significant, negative correlation between the two variables under study. This implies that while one variable rises, the other falls, and vice versa. The inverse relationship is indicated by the negative sign, and the strength of the association is indicated by the absolute value of 0.86. A correlation coefficient's range is from -1 to 1, where -1 denotes a perfect negative correlation, 0 denotes no correlation, and 1 denotes a perfect positive correlation. Thus, an r-value of -0.86 denotes a significant, negative correlation between the two variables.
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How many three-digit positive integers x are there, such that subtracting the sum of digits of x from x gives a three-digit number whose digits are all the same?
Answer:
Step-by-step explanation:Let $x$ be a three-digit positive integer, and let $a$, $b$, and $c$ be the digits of $x$. Then we want $x - (a + b + c)$ to be a three-digit number whose digits are all the same. This means that $x - (a + b + c) = 111$, $222$, $\ldots$, or $888$.
If $x - (a + b + c) = 111$, then we have $x = a + b + c + 111$. Since $x$ is a three-digit number, $1 \leq a \leq 9$, $0 \leq b \leq 9$, and $0 \leq c \leq 9$. Thus, there are $9 \cdot 10 \cdot 10 = 900$ possible values of $x$ that give a three-digit number whose digits are all $1$.
Similarly, we have $x = a + b + c + 222$, $x = a + b + c + 333$, $\ldots$, and $x = a + b + c + 888$. Each of these equations gives $900$ possible values of $x$.
Therefore, there are a total of $8 \cdot 900 = \boxed{7200}$ three-digit positive integers $x$ that satisfy the conditions of the problem.
A 30 microFarad capacitor is connected into a 240 V, 60 Hz circuit. What is the current flow in this circuit?
We must determine the voltage's rate of change with regard to time. The voltage changes sinusoidally with time in an AC circuit. the current flow in the circuit is [tex]3.817[/tex] Amperes.
What is the current flow in the circuit?To calculate the current flow in the circuit, we need to use the formula:
[tex]I = C x dV/dt[/tex]
Where:
I is the current flow in the circuit (in Amperes)
C is the capacitance of the capacitor (in Farads)
dV/dt is the rate of change of voltage with respect to time (in Volts per second)
First, we need to calculate the rate of change of voltage with respect to time. In an AC circuit, the voltage varies sinusoidally with time. The formula for voltage in an AC circuit is:
[tex]V = Vm sin(ωt)[/tex]
Where:
Vm is the maximum voltage (in Volts)
ω is the angular frequency (in radians per second)
t is the time (in seconds)
We know that the voltage is 240 V and the frequency is 60 Hz. We can use these values to calculate Vm and ω:
[tex]Vm = √2 x Vrms = √2 x 240 = 339.4 V[/tex]
[tex]ω = 2πf = 2π x 60 = 377 rad/s[/tex]
Now, we can calculate [tex]dV/dt:[/tex]
[tex]dV/dt = ω Vm cos(ωt)[/tex]
At [tex]t = 0[/tex] (i.e. when the voltage is at its maximum), [tex]cos(ωt) = 1. Therefore, dV/dt = ω Vm = 127,234 V/s.[/tex]
Finally, we can calculate the current flow:
[tex]I = C x dV/dt = 30 x 10^-6 F x 127,234 V/s = 3.817 A[/tex]
Therefore, the current flow in the circuit is 3.817 Amperes.
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the half life of a drug in the bloodstream is 3 hours. if you took a 600mg dose how much of the drug will remain after one day.will remain after one day? i got 2mg please help
Say no to drugs. It is not good for the body.
A forklift operator has the task of moving 43 palettes from one location to another. If the most palettes he can move at one time are 8 what are the minimum number of trips he will have to make from one location to the other in order to move all the palettes?
The forklift operator can move at most 8 palettes at one time.
What is Forklift operator?A forklift operator is a person who operates a forklift, which is a type of powered industrial truck used to lift and move heavy materials over short distances. Forklift operators are commonly employed in warehouses, manufacturing plants, and other industrial settings where materials need to be moved and stored. They are responsible for loading and unloading trucks, transporting materials to and from storage areas, and positioning materials for storage or processing. Forklift operators must be trained and certified to operate the equipment safely and efficiently.
In the given question,
To move all 43 palettes, the operator needs to make at least 43/8 = 5.375 trips.
Since the operator cannot make a fraction of a trip, he will need to make at least 6 trips in order to move all 43 palettes.
On the first five trips, the operator will move 8 palettes each time, and on the final trip, he will move the remaining 3 palettes.
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Please help!! Correct answer gets brainliest!!
Graph B is correct and shows the rectangular opening.
Looking at the text the question, I notice that each unit means 1 inch. After looking at all the graphs, I found out that B was the correct answer because of the sides. The sides are five inches by two inches and it was the only graph that had these units.
—————
Please remember to revise this and make it in your own words if you want! I hope this helps you. -Doodle
—————
Answer: Option B.
Step-by-step explanation:
With the scale of one unit equalling one inch, the only answer option that meets the given requirement of having side lengths that are 5 inches and 2 inches is option B.
Answer A is 4 units by 2 units, Answer B is the needed 5 units by 2 units, Answer C is 7 units by 2 units, and Answer D is 6 units by 2 units. This leaves us with Answer B.
The textbook says c = (1,2,3,4)
but according to my solution in the picture below,
I got c = (1,2,2,2).
Where did I get wrong?
(Don't mind the Korean words!
This question is about matrix factorization.)
According to the information the mistake is in the calculation of the vector y. According to your solution, y = [3, 2, 1, 0], but it should actually be y = [2, 2, 2, 2].
How to find the misftake?To see why, let's go back to the definition of y. We have:
y = c - Ax
Plugging in the given values, we get:
y = [1, 2, 3, 4] - [[1, 1, 0, 0], [0, 1, 1, 0], [0, 0, 1, 1]] [x1, x2, x3, x4]
Simplifying this matrix multiplication, we get:
y = [1-x1-x2, 2-x2-x3, 3-x3-x4, 4-x4]
Now we need to solve for x1, x2, x3, and x4 such that y = [2, 2, 2, 2]. That means we have the following system of equations:
1 - x1 - x2 = 2
2 - x2 - x3 = 2
3 - x3 - x4 = 2
4 - x4 = 2
Solving this system, we get x1 = -1, x2 = -1, x3 = 0, and x4 = 2. Plugging these values back into the equation for y, we get y = [2, 2, 2, 2], which is the correct solution.
So it looks like the mistake was in the calculation of y, which led to incorrect values for x1, x2, x3, and x4, and ultimately an incorrect value for c.
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How do I graph [tex]5\sqrt{y}[/tex] and [tex]y[/tex]. I know it's supposed to look like the picture below but I don't knw how to get there. How do I come up with the points and graph into the calculator.
Simply enter the equations into the calculator and use the graphing option to visualize these equations.
what is equations ?A mathematical assertion that two expressions are equal is known as an equation. One or more variables, or values that can change within a specific range, are frequently present in equations. The equal sign (=) is frequently used in equations to denote that the expressions on each side of it have the same value. For instance, the formula x + 2 = 5 states that the product of x and 2 equals 5. When we solve for x in this equation, we get x = 3, which is the value of x that causes the equation to hold true.
given
Follow these steps to graph y = 2x - 1 and y = -x + 3:
To plot, pick a set of x-values. For instance, you could select -5 x 5.
For each equation, enter x values to determine the associated y values. You can create a table to aid with work organization:
Draw a coordinate plane and plot the points from your table there. For instance, the first row of the table is represented by the point (-5, -11).
For each equation, draw a straight line to join the points. The slope of the line for y = 2x - 1 will be positive, whereas the slope of the line for y = -x + 3 will be negative.
Use a graphing calculator to double-check your work and make sure your graph matches the one in the image.
Simply enter the equations into the calculator and use the graphing option to visualize these equations.
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Circle A has center of (6, 7) and a radius of 4, and circle B has a center of (2, 4) and a radius of 16. What steps will help show that circle A is similar to circle B? Group of answer choices Translate circle A using the rule (x + 4, y + 3). Rotate circle A 45° about the center. Dilate circle A by a scale factor of 4. Reflect circle A about the origin.
Circle A can be made similar to circle B by dilating circle A by scale factor 4.
Circles can be made similar by performing transformations such as dilations, translation, etc.
The given parameters are:
Circle A
[tex](x,y)=(6,7)[/tex] --- center
[tex]r=4[/tex] --- radius
Circle B
[tex](x,y)=(2,4)[/tex] --- center
[tex]r=16[/tex] --- radius
To do this, we simply make both circles have the same radius.
Given that:
[tex]r_A=4[/tex] --- radius of A
[tex]r_B=16[/tex] --- radius of B
Dilating the radius of A by 4 will make both circles have the same radius.
i.e.
[tex]R=r_A\times4[/tex]
[tex]R=4\times4[/tex]
[tex]R=16[/tex]
Now, the new radius of A is 16; same as B
Both circles now have the same radius, means that the circles are similar.
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Suppose a ball is being twirled at the end of a string and the center of rotation is the origin of a coordinate system. If the string breaks, the initial path of the ball is on a line that is perpendicular to the radius of the circle. Suppose the string breaks when the ball is at the point whose coordinates are P(1, 2). Find the equation of the line on which the initial path lies.
The equation of the line on which the initial path of the ball lies is y = (-1/2)x + 5/2.
How to determine the equation of the line on which the initial path liesLet's call the point where the ball is initially released from the string "Q".
Since the ball is being twirled at the end of the string, its initial path will be tangent to the circle at point Q, and this tangent line will be perpendicular to the radius that passes through Q.
Since the center of rotation is the origin of the coordinate system, the radius of the circle passes through the origin and point Q. The equation of the radius can be found using the slope-intercept form:
slope of the radius = (y-coordinate of Q - y-coordinate of the origin) / (x-coordinate of Q - x-coordinate of the origin)
= (2 - 0) / (1 - 0) = 2
The radius also passes through the origin, so we can use the point-slope form of the equation:
y - y-coordinate of origin = slope of the radius * (x - x-coordinate of origin)
y - 0 = 2(x - 0)
y = 2x
Since the initial path of the ball is perpendicular to the radius at point P(1,2), its slope will be the negative reciprocal of the slope of the radius. The slope of the radius is 2, so the slope of the initial path is -1/2.
Using the point-slope form of the equation, we can find the equation of the line:
y - y-coordinate of P = slope of the initial path * (x - x-coordinate of P)
y - 2 = (-1/2)(x - 1)
y = (-1/2)x + 5/2
Therefore, the equation of the line on which the initial path of the ball lies is y = (-1/2)x + 5/2.
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If mWC = 108° and m/U = 37°, find mLQ.
The measure of arc LQ (m∠LQ) is equal to 34 degrees.
What is a chord of circleIn Mathematics and Geometry, a chord of circle can be defined as a line segment that typically join any two (2) points of a circle.
Generally speaking, the inscribed angle of a circle is equal to half of the central angle of a circle. Mathematically, this is represented by the following mathematical expression:
m∠U = 1/2(m∠WC - m∠LQ)
37 = 1/2(108 - m∠LQ)
74 = 108 - m∠LQ
m∠LQ = 108 - 74
m∠LQ = 34 degrees.
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Result of 6³⁶ pls i Need this
We know that we would need to make use of the calculator to obtain the value of 6³⁶ as 1.03 * 10^28.
What is the meaning of 6³⁶?The expression 6³⁶ means "6 raised to the power of 36," which is equivalent to multiplying 6 by itself 36 times.
So, 6³⁶ is a very large number. This number is difficult to comprehend, but it has many practical applications in fields such as physics, engineering, and computer science.
We can see that we have to make use of the standard form to write; 1.03 * 10^28 as the result.
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whats the answer also explain why its the answer
Answer:
B. Socialism
Step-by-step explanation:
In contrast to capitalism, socialists believe the shared ownership of resources and central planning offer a more equitable distribution of goods and services.
Karl Marx was the most prominent voice of socialism and believed the working class would rise up against the wealthy when faced with injustices.
7. The coordinates of the vertices of a hyperbola are (1, -1) and (1, 3).
The coordinates of the co-vertices of the same hyperbola are (-3, 1) and (5, 1).
Co-vertek
6 5 4 3 2 -1
b.
2
Vertex
4
(x+1)² (y+1)²
4
16
RET
Vertex
What would the standard form of the equation that describes the hyperbola?
(x-1)² (x-1)²
(y−1)²
16
4
(y+1)²
16
= 1
D
Co-vertex
✓
d.
(x−1)²
16
(x + 1)²
The answer is A. (x-1)²/16-(y-1)²/4=1. The standard form of the equation that describes the hyperbola is (x-1)²/16-(y-1)²/4=1.
What is hyperbola?A hyperbola is a two-dimensional, closed-curve shape with two branches, each of which is a mirror image of the other.
This equation is in the standard form of a hyperbola equation, which is
(x-h)²/a² - (y-k)²/b² = 1, where (h,k) is the center of the hyperbola and a and b are the distance from the center to the vertices of the hyperbola.
In this case, the center of the hyperbola is (1,1).
This can be determined by taking the average of the coordinates of the two vertices, (1,-1) and (1,3), which is (1,1).
The distance from the center to the vertices of the hyperbola is 8 for the x-coordinate, and 4 for the y-coordinate.
This can be determined by taking the difference of the coordinates of the vertices, which is 8 for the x-coordinate and 4 for the y-coordinate.
Therefore, the standard form of the equation that describes the hyperbola is (x-1)²/16-(y-1)²/4=1.
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Find the exact values of cos 150 and sin 150
50 points + brainiest if you answer fast
Match the following scale factors to the type of dilation that will occur:
Answer:
1. E, 2. C, 3. A, 4. D
Step-by-step explanation:
The negative sign means a that the image is on the opposite side.
The positive sign means a that the image is on the same side.
The magnitude of any factor between [0,1[ means that the image is contracted.
The magnitude of any factor equal to 1 means that the image is not dilated.
The magnitude of any factor between ]1,infinity[ means that the image is expanded.
can you find the limit?
Answer:
3
Step-by-step explanation:
what will be the cost of a carpeting a rectangular room of 6 m long and 4 m wide with the carpet of width 1.5 m at rupees 200 per metre
need explanation pls....
(>0<；)
To determine the cost of carpeting the room, we first need to calculate the area of the room that needs to be carpeted.
The area of the room is:
A = l × w
A = 6 m × 4 m
A = 24 square meters
Next, we need to determine the area of the carpet required. Since the carpet has a width of 1.5 m, we can think of the room as being covered by a series of strips of carpet, each of width 1.5 m. We need to calculate the length of each strip and then multiply that by the width of the carpet.
The length of each strip is equal to the length of the room, which is 6 m. So, the area of each strip of carpet is:
A_strip = l × w_carpet
A_strip = 6 m × 1.5 m
A_strip = 9 square meters
To cover the entire room, we will need to use several strips of carpet. The number of strips required is the total area of the room divided by the area of each strip:
Number of strips = A / A_strip
Number of strips = 24 square meters / 9 square meters
Number of strips = 2.6667 strips
Since we cannot use a fractional number of strips, we will need to round up to 3 strips of carpet.
The total area of carpet required is the area of each strip multiplied by the number of strips:
Total area of carpet = A_strip × Number of strips
Total area of carpet = 9 square meters × 3
Total area of carpet = 27 square meters
Finally, we can calculate the cost of carpeting the room at the given rate of rupees 200 per meter:
Cost of carpeting = Total area of carpet × Cost per meter
Cost of carpeting = 27 square meters × Rs. 200 per meter
Cost of carpeting = Rs. 5,400
Therefore, the cost of carpeting the rectangular room of 6 m long and 4 m wide with the carpet of width 1.5 m at rupees 200 per meter would be Rs. 5,400.
A second figure with a height of 19 ft is similar to the figure below. Find the surface area
and volume of the second figure.
7 ft
5 ft
6 ft
5 ft
Therefore, the surface area of the second figure is 3049.6 square feet, and its volume is 11,483.68 cubic feet.
What is volume?Volume is a measure of the amount of space that a three-dimensional object occupies. It is the measure of how much space is enclosed within a boundary in three dimensions. In simple terms, volume is the amount of space that an object occupies. It is measured in cubic units such as cubic meters (m³), cubic centimeters (cm³), cubic feet (ft³), or gallons (gal).
Here,
To find the surface area and volume of the second figure, we need to know the dimensions of the figure. Since the second figure is similar to the first one, we know that the ratios of the corresponding sides of the two figures are equal. Let x be the unknown scale factor between the two figures. Then, we have:
x = 19/5 (the ratio of the heights of the two figures)
So, we can find the other dimensions of the second figure by multiplying the corresponding dimensions of the first figure by x.
The dimensions of the first figure are:
Height = 5 ft
Length = 6 ft
Width = 7 ft
Multiplying these dimensions by x, we get:
Height = 19 ft (given)
Length = 6 * (19/5) = 22.8 ft
Width = 7 * (19/5) = 26.6 ft
Now, we can find the surface area and volume of the second figure.
Surface area = 2lw + 2lh + 2wh
= 2(22.8 * 26.6) + 2(22.8 * 19) + 2(26.6 * 19)
= 3049.6 ft²
Volume = lwh
= 22.8 * 19 * 26.6
= 11,483.68 ft³
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Complete question:
A second figure with a height of 19 ft is similar to the figure below. Find the surface area and volume of the second figure.
1. (5 points) What could be the areas of the three sails? Complete the table by giving a possible base, height, and area for all three sails.
Base length Height Area
Sail A
6 m
[between 5 and 8 m]
Sail B
[between Sail A and Sail B]
Sail C
[less than twice the area of Sail A]
The smallest sail, A, has a base length of 6 m and a height range of 5 to 8 m. The largest sail, sail C, has a smaller area than double that of sail A.
The table of possible dimensions and areas of the sails is:
Base Height Area
Sail A 6 m 5 m 15 m²
Sail B 6.14 m 7 m 21.5 m²
Sail C 14 m 4 m 28 m²
Let's assume that sail A has a height of 5 meters, then its area would be:
Area of Sail A = 1/2 x Base x Height = 1/2 x 6 x 5 = 15 square meters
If sail C has an area less than twice the area of sail A, then its area must be less than 2 x 15 = 30 square meters. Let's assume that sail C has an area of 28 square meters.
To find the dimensions of sail A and sail C, we can use the formula for the area of a triangle:
Area = 1/2 x Base x Height
For sail A, we already know the base and the area, so we can solve for the height:
15 = 1/2 x 6 x Height
Height = 5 meters
So sail A has a base of 6 meters and a height of 5 meters.
For sail C, we know the area and we have assumed a base length of 14 meters (to keep the height reasonable). Solving for the height:
28 = 1/2 x 14 x Height
Height = 4 meters
So sail C has a base of 14 meters and a height of 4 meters.
To find the dimensions of sail B, we can assume a height between 5 and 8 meters (since sail A has a height between 5 and 8 meters), and then solve for the base and the area using the formula for the area of a triangle:
Area = 1/2 x Base x Height
Let's assume sail B has a height of 7 meters:
Area of Sail B = 1/2 x Base x 7
Area of Sail B = 1/2 x (Area of Sail A + Area of Sail C)
Area of Sail B = 1/2 x (15 + 28)
Area of Sail B = 21.5 square meters
Solving for the base:
21.5 = 1/2 x Base x 7
Base = 6.14 meters (rounded to two decimal places)
So sail B has a base of 6.14 meters and a height of 7 meters.
Therefore, the table of possible dimensions and areas of the sails is:
Base Height Area
Sail A 6 m 5 m 15 m²
Sail B 6.14 m 7 m 21.5 m²
Sail C 14 m 4 m 28 m²
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PLEASE HELP ASAPPPPPPP
Answer: length is 325 and width is 137
Step-by-step explanation:
924=2L+2w
L= 188+w
924=2(188+w)+2w
924=376+2w+2w
924=376+4w
w=137
plug back into equation
L=188+w
^^ w=137
L=188+137
L=325
Quadrilateral ABCD has vertices A(2,4), B(5, 4), C(8, 1), and D(5, 1). Graph the image of quadrilateral ABCD after a dilation centered at D with a scale factor of
followed by a reflection in the line x = 1.
The final image of quadrilateral ABCD after a dilation centered at D with a scale factor of 2, followed by a reflection in the line x = 1, is attached below, where A', B', C', and D' are the new points after the transformation.
What are Transformation and Reflection?
Single or multiple changes in a geometrical shape or figure are called Geometrical Transformation. A geometrical transformation in which a geometrical figure changes his position to his mirror image about some point or line or axis is called Reflection.
First, we need to apply the dilation centered at point D with a scale factor of 2. To do this, we will multiply the coordinates of each point by 2, with D as the center of dilation.
Point D remains the same since the scale factor does not affect it:
D(5, 1)
To find the new coordinates of points A, B, and C, we will use the following formula:
New Point = Center of Dilation + Scale Factor * (Original Point - Center of Dilation)
For point A:
New A = D + 2 * (A - D)
New A = (5, 1) + 2 * (-3, 3)
New A = (-1, 7)
For point B:
New B = D + 2 * (B - D)
New B = (5, 1) + 2 * (0, 3)
New B = (5, 7)
For point C:
New C = D + 2 * (C - D)
New C = (5, 1) + 2 * (3, -3)
New C = (11, -5)
Now that we have the new coordinates, we can reflect the image in the line x = 1. To do this, we will reflect each point across the line x = 1 by replacing the x-coordinate with its reflection across the line.
The equation of line x = 1 is a vertical line passing through x = 1. Therefore, to reflect a point across this line, we need to find the difference between the x-coordinate of the point and the line (which is 1), and subtract that difference from the line.
For point D:
D(5, 1) reflected in x = 1 is (2, 1)
For point A:
New A(-1, 7) reflected in x = 1 is (-3, 7)
For point B:
New B(5, 7) reflected in x = 1 is (-3, 7)
For point C:
New C(11, -5) reflected in x = 1 is (13, -5)
Hence, The final image of quadrilateral ABCD after a dilation centered at D with a scale factor of 2, followed by a reflection in the line x = 1, is attached below, where A', B', C', and D' are the new points after the transformation.
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On the augmented matrix A below, perform all three-row operations in the order given, ((a) followed by (b) followed by (c)) and then write the resulting augmented matrix.
After applying the row operations, the matrix is obtained as [tex]\left[\begin{array}{c c c | c}1 & 1 & 1 & 3\\0 & 1 & -6 & -4 \\0 & 0 & -13 & -8\end{array}\right] \\[/tex].
What is a matrix?
A matrix is a rectangular array or table with numbers or other objects arranged in rows and columns. Matrices is the plural version of matrix. The number of columns and rows is unlimited. Matrix operations include addition, scalar multiplication, multiplication, transposition, and many others.
The augmented matrix is given as -
[tex]\left[\begin{array}{c c c | c}1 & 1 & 1 & 3\\2 & 3 & -4 & 2 \\-1 & -4 & 4 & 1\end{array}\right][/tex]
The first row operation is -
[tex]R_2=-2r_1+r_2[/tex]
After applying the row operation the matrix will be -
[tex]\left[\begin{array}{c c c | c}1 & 1 & 1 & 3\\0 & 1 & -6 & -4 \\-1 & -4 & 4 & 1\end{array}\right] \\[/tex]
The second row operation is -
[tex]R_3=1r_1+r_3[/tex]
After applying the row operation the matrix will be -
[tex]\left[\begin{array}{c c c | c}1 & 1 & 1 & 3\\0 & 1 & -6 & -4 \\0 & -3 & 5 & 4\end{array}\right] \\[/tex]
The third row operation is -
[tex]R_3=3r_2+r_3[/tex]
After applying the row operation the matrix will be -
[tex]\left[\begin{array}{c c c | c}1 & 1 & 1 & 3\\0 & 1 & -6 & -4 \\0 & 0 & -13 & -8\end{array}\right] \\[/tex]
Therefore, the matrix is obtained as [tex]\left[\begin{array}{c c c | c}1 & 1 & 1 & 3\\0 & 1 & -6 & -4 \\0 & 0 & -13 & -8\end{array}\right] \\[/tex].
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En una sustracción si el minuendo aumenta en 128 unidades halla en cuanto debe aumentar el sustraendo para que la diferencia disminuya en 67 unidades
So, if the minuend increases by 128 units, then the subtrahend does not need to increase in order to decrease the difference by 67 units.
What is equation?An equation is a mathematical statement that indicates that two expressions are equal. It contains an equals sign (=) between two expressions, with one expression on each side of the equals sign. Equations can contain variables, constants, coefficients, and mathematical operations such as addition, subtraction, multiplication, and division. The goal in solving an equation is to determine the value of the variable that makes both sides of the equation equal. Equations are an important tool in mathematics and are used in various fields such as physics, engineering, finance, and economics to model and solve problems.
Here,
Let's denote the minuend by M, the subtrahend by S, and the difference by D.
In a subtraction, we have: D = M - S
If the minuend increases by 128 units, then the new minuend M' is given by: M' = M + 128
We want to find how much the subtrahend must increase, let's call it x, so that the difference decreases by 67 units. This means that the new difference D' is given by: D' = D - 67
Substituting the expressions for D, M, and M' in terms of S, we get:
D' = (M + 128) - (S + x) - 67
Simplifying this expression, we get:
D' = (M - S) + 61 - x
But we know that D = M - S, so we can substitute it in the above expression:
D' = D + 61 - x
Now, we can substitute the values we know:
D' = D - 67 (since we want the difference to decrease by 67 units)
Therefore, we get:
D - 67 = D + 61 - x
Simplifying this expression, we get:
x = 128 - 67 - 61
x = 0
In other words, the difference will automatically decrease by 67 units when the minuend increases by 128 units.
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Complete question:
In a subtraction, if the minuend increases by 128 units, find by how much the subtrahend must increase so that the difference decreases by 67 units.
What is the area of the triangle shown below if the value of h is 3/4 of the length of the side that is labeled in the figure?