So the value of the integral ∫ dx/√(1-2x^2) over the interval [-1/√2, 1/√2] is (1/√2)π.
The value of the integral ∫ dx/√(1-2x^2) is given by:
∫ (1/√(1-2x^2)) dx = (1/2) * arcsin(√2 * x) + C
where C is the constant of integration.
The value of the integral ∫ dx/√(1-2x^2) is equal to the integral of the function 1/√(1-2x^2) with respect to x. This is an example of an integral that requires a trigonometric substitution to evaluate. Specifically, we can let x = sin(θ)/√2 and dx = cos(θ)/√2 dθ. Substituting these expressions into the integral yields:
∫ dx/√(1-2x^2) = ∫ (cos(θ)/√2) / √(1-2(sin(θ)/√2)^2) dθ
Simplifying the denominator gives:
√(1-2(sin(θ)/√2)^2) = √(1 - sin^2(θ)) = cos(θ)
Substituting this expression into the integral gives:
∫ dx/√(1-2x^2) = ∫ (cos(θ)/√2) / cos(θ) dθ = ∫ dθ/√2 = (1/√2)θ + C
To find the value of the integral, we need to substitute back in for x and evaluate at the limits of integration. If we are integrating over the interval [-1/√2, 1/√2], then:
(1/√2)θ evaluated from -π/4 to π/4 gives:
(1/√2)(π/4 + π/4) - (1/√2)(-π/4 - π/4) = (1/√2)π
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On the same coordinate plane, mark all points (x,y) such that (A) y=x-2, (B) y=-x-2, (C) y=|x|-2
The marked points are (-2,-4), (-2,-1), (0,-2), (2,-1), and (2,-4), under the condition that they are on the same coordinate plane having (A) y=x-2, (B) y=-x-2, (C) y=|x|-2.
In the given graph points on the coordinate plane, we have to plot the points (x,y)
Here
x = horizontal axis
y = vertical axis.
In the given point A, y=x-2, we can continue at the origin (0,0) and move 2 units go down on the y-axis and 2 units right on the x-axis to plot point A at (2,0).
In the given point B, y=-x-2, we can continue at the origin (0,0) and transfer 2 units down on the y-axis and 2 units left on the x-axis to plot point B at (-2,0).
In the given point C, y=|x|-2, we can continue plotting two points for this equation.
When x is considered negative, we can procees at the origin (0,0) and transfer 2 units down on the y-axis and 2 units left on the x-axis to plot point C at (-2,0).
When x is positive, we can start at the origin (0,0) and move 2 units down on the y-axis and 2 units right on the x-axis to plot point C at (2,0).
Then, all points (x,y) such that (A) y=x-2, (B) y=-x-2, (C) y=|x|-2 are (-2,-4), (-2,-1), (0,-2), (2,-1), and (2,-4).
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If the equation is y = 2x^2 + 4x - 6 what is zero #1 (x,y) and zero #2 (x,y)
The zero #1 is (-3, 0) and the zero #2 is (1, 0).
To find the zeros of the quadratic equation y = 2x² + 4x - 6, we need to solve for the values of x when y = 0.
We can start by setting y to zero:
0 = 2x² + 4x - 6
Next, we can divide both sides by 2 to simplify the equation:
0 = x² + 2x - 3
We can then factor the left-hand side of the equation:
0 = (x + 3)(x - 1)
Using the zero product property, we can set each factor equal to zero and solve for x:
x + 3 = 0 or x - 1 = 0
x = -3 or x = 1
So the zeros of the quadratic function are (-3,0) and (1,0).
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PLEASE HELP ME IMMEDIATELY!!!!!
The intervals where f is decreasing are given as follows:
None of the above.
When a function is increasing and when it is decreasing, looking at it's graph?Looking at the graph, we get that a function f(x) is increasing when it is "moving northeast", that is, to the right and up on the graph, meaning that when the input variable represented x increases, the output variable represented by y also increases.Looking at the graph, we get that a function f(x) is decreasing when it is "moving southeast", that is, to the right and down the graph, meaning that when the input variable represented by x increases, the output variable represented by y decreases.Hence the decreasing intervals of the function are given as follows:
-3.5 < x < -1.x > 2.5.Which are none of the options given in the problem.
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Ben completely filled his 20-gallon tank of gas with regular fuel for $59. 80 as he left the gas station he noticed the gas station across the street sold regular fuel for $2. 84 a gallon how much money could ben have saved per gallon if he had gone to the gas station across the street
Ben could have saved $0.15 per gallon if he had gone to the gas station across the street.
If Ben filled his 20-gallon tank of gas with regular fuel for $59.80, then the cost per gallon can be found by dividing the total cost by the number of gallons:
cost per gallon = total cost / number of gallons
cost per gallon = $59.80 / 20 gallons
cost per gallon = $2.99/gallon
If the gas station across the street sold regular fuel for $2.84 a gallon, then the amount Ben could have saved per gallon is:
savings per gallon = cost per gallon at initial station - cost per gallon at other station
savings per gallon = $2.99/gallon - $2.84/gallon
savings per gallon = $0.15/gallon
Therefore, Ben could have saved $0.15 per gallon if he had gone to the gas station across the street.
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Find the area of the shaded region.
round to the nearest tenth.
1230
18.6 m
area = [ ? ] m2
The area of the shaded region is 422.8 m², rounded to the nearest tenth.
To find the area of the shaded region, we first need to determine the areas of the two shapes that make up the region. The first shape is a rectangle with dimensions of 18.6 m by 30 m, which has an area of:
Area of rectangle = length x width = 18.6 m x 30 m = 558 m²
The second shape is a semi-circle with a diameter of 18.6 m, which has a radius of 9.3 m. The area of a semi-circle is half the area of a full circle, so we can use the formula for the area of a circle to find the area of the semi-circle:
Area of semi-circle = (1/2) x π x r² = (1/2) x π x 9.3² = 135.2 m²
To find the area of the shaded region, we need to subtract the area of the semi-circle from the area of the rectangle:
Area of shaded region = Area of rectangle - Area of semi-circle
Area of shaded region = 558 m² - 135.2 m²
Area of shaded region = 422.8 m²
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Keegan deposited $675 in a savings account that pays 4.8% annual interest compounded quarterly.
Write the compound interest formula to represent Keegan's investment after 5 years.
How much money will Keegan have in the account after 5 years?
Keegan will have approximately $878.85 in the account after 5 years.
What is Compound interest ?
Compound interest is the interest that is earned not only on the initial amount of money invested (known as the principal), but also on any interest earned on that principal over time. In other words, compound interest is interest on interest.
The compound interest formula is given by:
A = P[tex](1 + r/n)^{nt}[/tex]
where A is the amount after t years, P is the principal amount, r is the annual interest rate, n is the number of times the interest is compounded per year, and t is the time in years.
In this case, Keegan deposited $675, the annual interest rate is 4.8%, the interest is compounded quarterly, and the investment is for 5 years. Therefore, we can plug in these values into the formula to get:
A = 675[tex](1 + 0.048/4)^{20}[/tex]
A = 675[tex](1.012)^{20}[/tex]
A ≈ $878.85
Therefore, Keegan will have approximately $878.85 in the account after 5 years.
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Write the equation in factored form:
so x= what values?
What do the solutions for x mean?
x2−4x−21=0
Assume the annual rate of change in the national debt of a country (in billions of dollars per year) can be modeled by the function
D'(t)=858.29+819.48t-184.32t^2+12.12t^3
where t is the number of years since 1995. By how much did the debt increase between 1996 and 2003 ?
The debt increased between 1996 and 2003. Then the national debt increased by approximately $4,903.73 billion between 1996 and 2003.
To find how much the debt increased between 1996 and 2003, we need to find the value of the function D'(t) for t=7 (since 2003 is 7 years after 1996).
D'(t)=858.29+819.48t-184.32t^2+12.12t^3
D'(7)=858.29+819.48(7)-184.32(7^2)+12.12(7^3)
D'(7)=858.29+5,736.36-8,132.32+3,458.68
D'(7)=1,921.01
Therefore, the annual rate of change in the national debt in 2003 was $1,921.01 billion per year.
To find how much the debt increased between 1996 and 2003, we need to integrate the function D'(t) from t=1 to t=7:
∫(D'(t))dt = ∫(858.29+819.48t-184.32t^2+12.12t^3)dt
= 858.29t + 409.74t^2 - 61.44t^3 + 3.03t^4 + C
where C is the constant of integration.
Evaluating this expression at t=7 and t=1 and taking the difference, we get:
(858.29(7) + 409.74(7)^2 - 61.44(7)^3 + 3.03(7)^4 + C) - (858.29(1) + 409.74(1)^2 - 61.44(1)^3 + 3.03(1)^4 + C)
= 6,111.09 - 1,207.36 = 4,903.73
Therefore, the national debt increased by approximately $4,903.73 billion between 1996 and 2003.
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The equation, with a restriction on x, is the terminal side of an angle theta in standard position. 5x + y = 0, x ≥ 0 Give the exact values of the six trigonometric functions of theta. Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. sin theta = (Simplify your answer. Use integers or fractions for any numbers in the expression. Type an exact answer, using radicals as needed. Rationalize all denominators.) B. The function is undefined.
Sin theta = -5/sqrt(26) Is the correct option , To find the six trigonometric functions of theta, we need to first find the values of x and y on the terminal side of theta. From the given equation, 5x + y = 0 and x ≥ 0, we can solve for y in terms of x:
y = -5x
Since x ≥ 0, we know that (x, y) lies in the fourth quadrant. We can now use the Pythagorean theorem to find the length of the hypotenuse:
r =[tex]sqrt(x^2 + y^2) = sqrt(x^2 + (-5x)^2) = sqrt(26x^2)[/tex]
Now we can find the six trigonometric functions:
sin(theta) = y/r = [tex]-5x/sqrt(26x^2) = -5/sqrt(26)[/tex]
cos(theta) = x/r =[tex]x/sqrt(26x^2) = 1/sqrt(26)[/tex]
tan(theta) = y/x = -5x/x = -5
csc(theta) = r/y = [tex]sqrt(26x^2)/(-5x) = -sqrt(26)/5[/tex]
sec(theta) = r/x = sqrt(26x^2)/x = sqrt(26)/1 = sqrt(26)
cot(theta) = 1/tan(theta) = -1/5
Therefore, the exact values of the six trigonometric functions of theta are:
sin(theta) = -5/sqrt(26)
cos(theta) = 1/sqrt(26)
tan(theta) = -5
csc(theta) = -sqrt(26)/5
sec(theta) = sqrt(26)
cot(theta) = -1/5
Answer: A. sin theta = -5/sqrt(26)
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Prove the following trigonometric identities
csc x - sin x - cosx cos x
To prove the trigonometric identity csc(x) - sin(x) - cos(x)cos(x), we will start by simplifying the left-hand side of the equation using trigonometric definitions and identities.
Recall that csc(x) = 1/sin(x). We will use this definition to rewrite the left-hand side of the equation:
1/sin(x) - sin(x) - cos(x)cos(x)
Now, we will find a common denominator for the terms in the equation. In this case, the common denominator is sin(x). To do this, we will multiply sin(x) to the second term:
(1 - sin^2(x) - cos(x)cos(x)sin(x)) / sin(x)
Next, we will use the Pythagorean identity sin^2(x) + cos^2(x) = 1 to replace sin^2(x) in the expression:
(1 - (1 - cos^2(x)) - cos(x)cos(x)sin(x)) / sin(x)
Simplifying the expression, we get:
(cos^2(x) - cos(x)cos(x)sin(x)) / sin(x)
Now, we can factor out cos(x) from the numerator:
cos(x)(cos(x) - sin(x)) / sin(x)
This expression is equivalent to the given identity, so we have proven the trigonometric identity:
csc(x) - sin(x) - cos(x)cos(x) = cos(x)(cos(x) - sin(x)) / sin(x)
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Help asap
if someone has the correct answer in the next 20 mins they will be awarded brainiest
insert a monomial so that the result is an identity.
(... - b 4)(64 + ....) = 121a 10 – 68
We can replace the ellipsis with -49a4 to get: (7/4)a 10 - 2 = -49a4 - b 4, missing monomial is -49a4.
To find the missing monomial, we need to use the distributive property to expand the left side of the equation:
(... - b 4)(64 + ....) = 121a 10 – 68
Expanding the left side gives:
64(... - b 4) + ....(... - b 4) = 121a 10 – 68
Now we need to find a monomial to replace the ellipsis that will make the left side of the equation equal to the right side of the equation, no matter what value is substituted for a and b.
Looking at the constants in the equation, we can see that 121 and 68 have a common factor of 17. Therefore, we can divide both sides of the equation by 17 to simplify the coefficients:
7a 10 - 4 = 4(... - b 4) + 4
Simplifying further, we get:
7a 10 - 8 = 4(... - b 4)
Dividing both sides by 4, we get:
(7/4)a 10 - 2 = ... - b 4
Now we need to find a monomial to replace the ellipsis that will make the left side of the equation equal to the right side of the equation, no matter what value is substituted for a and b.
Since we want the left side of the equation to have a degree of 4 (because the right side has a term of -b4), we need to choose a monomial of degree 4 that will cancel out the terms of degree 10 on the left side. One possible monomial is:
-49a4
Therefore, we can replace the ellipsis with -49a4 to get: (7/4)a 10 - 2 = -49a4 - b 4
So the missing monomial is -49a4.
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Evaluate the following expression. Your answer must be in exact form: for example, type pi/6 for π/6 or DNE if the expression is undefined. arcsin (sin (-57π/10))=
For the following expression arcsin (sin (-57π/10)) is 3π/10.
The function arcsin(x) gives the angle in radians whose sine is x.
In this problem, we need to find the angle whose sine is equal to the sine of -57π/10.
First, we need to simplify -57π/10 to an angle in the range [-π/2, π/2] since the sine function has a range of [-1, 1]. To do this, we use the fact that sine has a period of 2π,
which means that sin(-57π/10) = sin((-57π/10) + 4π) = sin(3π/10).
So we need to find the angle θ such that sin(θ) = sin(3π/10).
Since sine is an odd function, we know that sin(-θ) = -sin(θ), so we can also say that sin(θ) = sin(-3π/10).
Therefore, there are two possible angles that satisfy the equation: θ = 3π/10 or θ = -3π/10.
However, since the range of the arcsine function is [-π/2, π/2], only the angle in that range that satisfies the equation is θ = 3π/10.
Therefore, we can write:
arcsin(sin(-57π/10)) = arcsin(sin(3π/10)) = 3π/10
The answer is 3π/10.
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A local amusement park found that if the admission was $7, about 1000 customers per day were admitted. When the admission was dropped to $6, the park had about 1200 customers per day. Assuming a linear demand function, determine the admission price that will yield maximum revenue.
The admission price that will yield maximum revenue is $6.
To determine the admission price that will yield maximum revenue, we'll first find the linear demand function using the given data points: ($7, 1000) and ($6, 1200).
Let x represent the admission price and y represent the number of customers per day. We can calculate the slope (m) using the formula:
m = (y2 - y1) / (x2 - x1)
Substituting the given data points:
m = (1200 - 1000) / (6 - 7) = 200 / (-1) = -200
Now, we have the slope and a point, so we can use the point-slope form to find the linear demand function:
y - y1 = m(x - x1)
Using the point ($7, 1000):
y - 1000 = -200(x - 7)
Now, let's rewrite the equation to the slope-intercept form (y = mx + b):
y = -200x + 2400
The revenue (R) is equal to the product of the admission price (x) and the number of customers (y):
R = xy
Substitute the linear demand function (y = -200x + 2400) into the revenue equation:
R = x(-200x + 2400)
To maximize the revenue, we need to find the vertex of the parabola represented by this equation. The x-coordinate of the vertex is given by:
x_vertex = -b / 2a
In this case, a = -200 and b = 2400:
x_vertex = -2400 / (2 * -200) = 6
The admission price that will yield maximum revenue is $6.
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What is the length of the line?
A. 9
B. 8
C. squared 45
D. squared 27
Answer:
C) [tex]\sf \sqrt{45}[/tex]
Step-by-step explanation:
Pythagorean theorem:
AB = 6 units
BC = 3 units
AC is hypotenuse and AB is the base and BC is the altitude.
Hypotenuse² = base² + altitude²
AC² = AB² + BC²
[tex]\sf = 6^2 + 3^2\\\\ = 36 + 9\\\\ = 45[/tex]
[tex]\sf AC= \sqrt{45}[/tex]
Ted Pappas pays
$4,388. 65 in real estate taxes yearly. His property has a market value of
$119,340. 00 with a rate of assessmentof 42%. What is his tax rate to the
nearest tenth of a mill?
Ted Pappas' tax rate to the nearest tenth of a mill is approximately 87.6 mills.
First, let's determine the assessed value of the property. To do this, we'll multiply the market value by the rate of assessment:
Assessed Value = Market Value × Rate of Assessment
Assessed Value = $119,340 × 0.42
Assessed Value = $50,122.80
Now, we need to find the tax rate in mills. One mill is equal to $1 per $1,000 of assessed value. To find the tax rate, we'll divide the yearly real estate taxes by the assessed value and multiply by 1,000:
Tax Rate (in mills) = (Yearly Real Estate Taxes / Assessed Value) × 1,000
Tax Rate = ($4,388.65 / $50,122.80) × 1,000
Tax Rate ≈ 87.6 mills
Therefore, Ted Pappas' tax rate to the nearest tenth of a mill is approximately 87.6 mills.
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Please help asap!! the image is attached, 35 points!!
Answer:
B
Step-by-step explanation:
Lisa's first step is to distribute, so you have to distribute all numbers.
(3+6x)-2(x+1)+5
3(1+2x)-2)x+1)+5
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Complete the following to use the difference of two squares to find the product of 22 and 18.( + )( - ) =( )2 - ( )2 =396
The complete equation of two squares to find the product of 22 and 18 is (22 + 18)(22 - 18) = 396
When we can interpret an expression as the difference of two perfect squares, i.e. a2-b2, we can factor it as (a+b)(a-b).
To use the difference of two squares to find the product of 22 and 18:
First, find the average of the two numbers:
(22 + 18) ÷ 2 = 20
Then, find the difference between the two numbers:
22 - 18 = 4
Now we can write:
(20 + 4)(20 - 4) = 24 × 16 = 384
But we need to add the extra 12 to get 396:
(20 + 4)(20 - 4) + 12 = 396
So the completed equation is:
(22 + 18)(22 - 18) = 396
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Jess has $43 in her bank account. Destiny is negative 21. What's the difference between Jess and Destiny's account?
Answer:
$64
Step-by-step explanation:
This would be the answer because if we figure out what needs to be added to -21 to reach 43 we get 64.
(-21+64 - $43.)
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1/2 (7)(4) + 6(5)=
I can not figure this out! Can you answer with middle school techniques?
The value of the given expression is 44. The solution has been obtained by using the arithmetic operations.
What are arithmetic operations?
The four basic operations, also referred to as "arithmetic operations," are thought to explain all real numbers. Operations like division, multiplication, addition, and subtraction come before operations like quotient, product, sum, and difference in mathematics.
We are given an expression as [tex]\frac{1}{2}[/tex] (7) (4) + 6 (5).
We know that when there is no sign in between two numbers, it denotes multiplication.
So, we get
⇒ [tex]\frac{1}{2}[/tex] * (7) * (4) + 6 * (5)
⇒ 14 + 30
⇒ 44 (Using addition operation)
Hence, the value of the given expression is 44.
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Consider the following. u = (5, -9, -5), v = (-7, -4, 3) (a) Find the projection of u onto v
Projection of vector u onto vector v is approximately (1.33, 0.76, -0.57).
How to find the projection of vector u onto vector v?We'll use the following formula:
projection of u onto v = (u•v / ||v||²) * v
First, we need to calculate the dot product (u•v) and the magnitude squared (||v||²) of vector v.
1. Dot product (u•v):
u•v = (5 * -7) + (-9 * -4) + (-5 * 3) = -35 + 36 - 15 = -14
2. Magnitude squared (||v||²):
||v||^2 = (-7)² + (-4)² + (3)² = 49 + 16 + 9 = 74
Now, we'll plug these values into the projection formula:
projection of u onto v = (-14 / 74) * v
We'll multiply each component of vector v by the scalar (-14/74):
projection of u onto v = (-14/74) * (-7, -4, 3) = (1.33, 0.76, -0.57)
So, the projection of vector u onto vector v is approximately (1.33, 0.76, -0.57).
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Find x.
Please help thank you
2. A particular ostrich runs 40 miles per hour. Select the animals who run at a faster unit rate per hour than the
ostrich. Mark all that apply.
A. O giraffe: 96 miles in 3 hours
B. O elk: 90 miles in 2 hours
c. O lion: 150 miles in 3 hours
D. O squirrel: 36 miles in 3 hours
A teacher conducts a survey of 50 randomly selected 6th grade and 50 randomly selected 7th grade students. The survey asks the students about the type of books they like to read. The table shows the number of students who selected each type of book
In a school, a teacher surveys 50 randomly selected 6th grade students and 50 randomly selected 7th grade students. c) More seventh-graders than sixth-graders enjoy horror films.
The first statement is false as the total 6th graders which like horror and comedy movie is 19 + 9 = 28 students which is more than 6th graders who like action movies which is 22, hence the first statement is false. this is interpreted from given data set.
The second statement is also false as it says that 6th graders prefer comedy films to action films, whereas 7th graders prefer action films but from the data given, it can be seen that the number of 6th graders who like comedy films is same as the number of 7th graders who like action movies which is 19, hence statement is false.
The third statement is true as 6th graders who like horror movies is 9 while 7th graders who like horror movies is 14 and hence, the statement is true.
Fourth statement is also false as 17, 7th graders like comedy movies in contrast to 14, 7th graders who like horror movies.
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Correct question:
A teacher conducts a survey of 50 randomly selected 6th grade and 50 randomly selected 7th grade students in a school. The survey asks the students about the type of movies they like to watch. The table shows the number of students who selected each type of movie. Select the correct statement.
a) 6th graders like action movies more than horror and comedy movies.
b) 6th graders like comedy movies more than 7th graders like action movies.
c) More 7th graders than 6th graders like horror movies.
d) 7th graders like horror movies more than comedy movies.
Find the magnitude of v. v = 7i
|lv|| = _____
The magnitude of vector v is:
|v| = 7
The magnitude of v is simply the length of the vector v, which can be found using the Pythagorean theorem. The vector v is given as v = 7i.
To find the magnitude of v (|v|), use the formula:
|v| = √(x² + y²)
where x and y are the components of the vector v. In this case, x = 7 (from 7i) and y = 0 (since there is no j component).
Now, plug in the values of x and y into the formula:
|v| = √(7² + 0²)
|v| = √(49 + 0)
|v| = √(49)
|v| = 7
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The sides of a triangle have lengths
5, 7 and x.
a. For what values of x is the
triangle a right triangle?
b. Tell whether the side lengths
form a Pythagorean triple.
(a) The triangle is a right triangle when x is equal to 24.
(b) The side lengths do not form a Pythagorean triple.
(a) To determine when the triangle is a right triangle, we can apply the Pythagorean theorem, which states that in a right triangle, the square of the length of the hypotenuse (the longest side) is equal to the sum of the squares of the lengths of the other two sides.
Using the given side lengths, we have:
5^2 + 7^2 = x^2
Simplifying the equation:
25 + 49 = x^2
74 = x^2
To find the value of x, we take the square root of both sides:
x = √74
Approximating the square root of 74, we get:
x ≈ 8.60
Therefore, when x is approximately 8.60, the triangle is a right triangle.
(b) For the side lengths to form a Pythagorean triple, they must satisfy the condition of the Pythagorean theorem. In this case, we have:
5^2 + 7^2 = x^2
Simplifying the equation:
25 + 49 = x^2
74 = x^2
Since the sum of the squares of the two smaller sides (25 + 49 = 74) is not equal to the square of the longest side (x^2 = 74), the side lengths of 5, 7, and x do not form a Pythagorean triple.
In conclusion, the triangle is a right triangle when x is equal to approximately 8.60, and the side lengths do not form a Pythagorean triple.
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Baron Blakk decided to start saving. He deposited seven thousand kroner at the start of each year. He always received a fixed interest rate of 2.3%. How much interest had he received in total after eight years? Give the answer to two decimal places.
The interest he had received in total after eight years is 1396.59 kroner
How much interest had he received in total after eight years?From the question, we have the following parameters that can be used in our computation:
Principal, P = 7000
Rate, r = 2.3%
Time, t = 8
The interest he had received in total after eight years is calculated as
I = P(1 + r)^t - P
Substitute the known values in the above equation, so, we have the following representation
I = 7000(1 + 2.3%)^8 - 7000
Evaluate
I = 1396.59
Hence, the total interest is 1396.59 kroner
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Graph
y=4x−9y, equals, 4, x, minus, 9.
The line crosses the y-axis at -9 and has a slope of 4, meaning it increases 4 units in the y-direction for every 1 unit increase in the x-direction.
What is graph?A graph is a visual representation of data, typically involving the use of points, lines, and curves to show how different values are related to each other. Graphs can be used to display a wide range of information, from mathematical functions and scientific data to business trends and social networks.
Here,
To graph y = 4x - 9, we can start by plotting the y-intercept, which is -9 on the y-axis. Then, we can use the slope of 4 to find other points on the line.
Slope = 4 can be written as rise/run = 4/1. This means that for every increase of 1 in the x-direction, the y-value increases by 4.
Starting from the y-intercept (-9,0), we can go up 4 units and to the right 1 unit to get the point (1,-5). We can continue this pattern and plot more points on the line, or we can simply draw a straight line through the y-intercept and (1,-5) to represent the equation y = 4x - 9.
Here's what the graph looks like:
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The lengths of the sides of a triangle are given. Classify each triangle as acute,
right, or obtuse.
19. 3, 4, 6
To start, compare c 2 to a 2 + b2
. Substitute the greatest length for c.
20. 9, 11, 16
19. The triangle with sides 3, 4, 6 is obtuse triangle.
20. The triangle with sides 9, 11, 16 is acute triangle.
How to Classify Triangles?In a triangle, the sum of the squares of the two shorter sides must be greater than the square of the longest side for the triangle to be considered "non-degenerate," which means it is a valid triangle with a positive area.
19. We have sides of 3, 4, and 6. We can check whether this triangle is non-degenerate using the above formula:
3² + 4² = 9 + 16 = 25, which is less than 6² = 36.
Therefore, this triangle is non-degenerate and we can classify it based on the size of its angles.
To do so, we can use the Pythagorean Theorem to find that the longest side (6) is opposite the largest angle, which is obtuse. Therefore, triangle #19 is an obtuse triangle.
20. For triangle #20, we have sides of 9, 11, and 16. Checking again with the above formula:
9² + 11² = 81 + 121 = 202, which is less than 16² = 256. So this triangle is also non-degenerate.
Using the same method as before, we can find that the longest side (16) is opposite the largest angle, which is acute. Therefore, triangle #20 is an acute triangle.
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For the function f(x)= 4x³ – 36x² +1.
(a) Find the critical numbers of f(if any) (b) Find the open intervals where the function is increasing or decreasing.
(a) The critical numbers are x = 0 and x = 6.
(b) The function is increasing on the interval (0, 6) and decreasing on the intervals (-∞, 0) and (6, ∞).
How to find the critical numbers of f(x)?(a) To find the critical numbers of f(x), we need to find the values of x where f'(x) = 0 or f'(x) does not exist.
f'(x) = 12x² - 72x
Setting f'(x) = 0, we get:
12x² - 72x = 0
12x(x - 6) = 0
So, the critical numbers are x = 0 and x = 6.
How to determine where the function is increasing or decreasing?(b) To determine where the function is increasing or decreasing, we need to examine the sign of f'(x) on different intervals.
For x < 0, f'(x) = 12x² - 72x < 0, which means the function is decreasing on (-∞, 0).
For 0 < x < 6, f'(x) = 12x² - 72x > 0, which means the function is increasing on (0, 6).
For x > 6, f'(x) = 12x² - 72x < 0, which means the function is decreasing on (6, ∞).
So, the function is increasing on the interval (0, 6) and decreasing on the intervals (-∞, 0) and (6, ∞).
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Given the sequence an = n^7/n^4
The sequence is = _______
Given the sequence an = n^7/n^4, The sequence is = divergent. So, the answer is that the sequence is divergent.
The sequence an = n^7/n^4 can be simplified by cancelling out the common factor of n^4 in the numerator and denominator. This yields an = n^3.
Therefore, the sequence is simply the cubes of the positive integers, or 1, 8, 27, 64, 125, 216, ... and so on. This sequence grows without bound, as n gets larger, so it does not converge to any finite limit. In other words, the sequence an does not have a limit, and it is divergent.
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