Answer:
1
Step-by-step explanation:
1/3 + 8/12 ⬅️(factor out by common factor, 4)
= 1/3 + 2/3
=1+2/ 3
=3/3 or 1
Find an equation of the plane.
The plane that passes through the line of intersection of the planes
x − z = 3 and y + 3z = 3
and is perpendicular to the plane
x + y − 4z = 6
The equation of the plane that passes through the line of intersection of x - z = 3 and y + 3z = 3 and is perpendicular to x + y - 4z = 6 is x + y - 4z = 3.
What is point normal form?The point-normal form of the equation of a plane is given by:
N · (<x - x0>, <y - y0>, <z - z0>) = 0
Where (x0, y0, z0) is a point on the plane and N = is a normal vector to the plane, we have the point-normal form of the equation of a plane. The dot product of the vector from the supplied location to any point on the plane with the normal vector to the plane yields this form of the equation. The equation states that any vector located in the plane with the normal vector has a zero dot product. The scalar equation of the plane can also be found by expanding the dot product, and it takes the form axe + by + cz = d, where d = N (x0, y0, z0).
Given the equation of the planes is x − z = 3 and y + 3z = 3.
Now, find the direction vector of the line of intersection:
Set z = t:
x = t + 3 and y = 3 - 3t
The direction vector is <1, -3, 1>.
2. Determine the normal vector:
The plane is perpendicular to the plane x + y - 4z = 6, so:
normal vector of x + y - 4z = 6, which is <1, 1, -4>.
3. Using point normal form we have:
(3, 0, 0)
The point satisfies the equation:
x - z = 3 and y + 3z = 3 when z = 0
Thus,
<1, 1, -4> · <x - 3, y, z> = 0
x + y - 4z = 3
Hence, the equation of the plane that passes through the line of intersection of x - z = 3 and y + 3z = 3 and is perpendicular to x + y - 4z = 6 is x + y - 4z = 3.
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solve y''+y=t using laplace inverse with y(0)=1 and y'(0)=-2
The solution of the differential equation y'' + y = t with the initial conditions y(0)=1 and y'(0)=-2 is y(t)= 1-2t+te-t.
What is equation?Equation is a mathematical statement that expresses the equality of two expressions. It shows the relationship between two or more variables and can be written using symbols, numbers, and operations. Equations are used to describe physical laws, to make calculations, and to solve problems. Examples of equations include the Pythagorean theorem, Newton's laws of motion, and linear equations.
We solve this differential equation using Laplace inverse, with the initial conditions y(0)=1 and y'(0)=-2. First, we take the Laplace transform of the equation:
L[y''+y]=L[t]
Using the properties of Laplace transform, we can write this as:
s2Y(s)-sy(0)-y'(0)+Y(s)= (1/s)
Substituting the initial conditions and rearranging terms, we have:
Y(s)= (1/s) + (2/s2) + (1/s2)
We can then invert the Laplace transform to get the solution of the original equation:
y(t)= 1-2t+te-t
Therefore, the solution of the differential equation y'' + y = t with the initial conditions y(0)=1 and y'(0)=-2 is y(t)= 1-2t+te-t.
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Determine the circumference and approximate area of the
given circle, using 3.14 for pie.
The circumference and approximate area of the given circle is 69.08 inches & 380.14 square inches.
What is circumference?
Circumference is the distance around the edge of a circular object or a round shape. It is the length of the boundary or perimeter of the circle. The formula is given by C = 2πr, where C is the circumference, r is the radius of the circle, and π is a mathematical constant approximately equal to 3.14.
The circumference of a circle is given by the formula:
C = 2πr
where r is the radius of the circle and π (pi) is a mathematical constant approximately equal to 3.14.
Using this formula and plugging in the given value of radius:
C = 2 x 3.14 x 11
C = 69.08 inches (rounded to two decimal places)
So the circumference of the circle with 11 inches radius is approximately 69.08 inches.
The area of a circle is given by the formula:
A = πr²
Again, using the given value of radius and approximating π to 3.14:
A = 3.14 x 11²
A = 3.14 x 121
A = 380.14 square inches (rounded to two decimal places)
So the approximate area of the circle with 11 inches radius is approximately 380.14 square inches.
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Maria works for an online auto trader. She makes a piecewise function to show the cost to place an online
advertisement.
(39
(39+5(x-6)
What is the cusp of the function?
c(x)
whenx ≤6
when x>6
According to the given information, the function has no cusp.
What is a function?
A function is a relation between a set of inputs and a set of possible outputs, with the property that each input is related to exactly one output.
The given piecewise function is:
c(x) = 39, when x ≤ 6
c(x) = 39 + 5(x - 6), when x > 6
A cusp is a point on the graph where the function changes direction very abruptly, like a sharp turn. This happens when the derivative of the function is not defined at that point.
The derivative of the function is:
c'(x) = 0, when x ≤ 6
c'(x) = 5, when x > 6
Since the derivative is defined and continuous at x = 6, there is no cusp at that point. Therefore, the function has no cusp.
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Find an equation of the osculating plane and an equation of the normal
plane of the curve x = sin 2t, y = t, z = cos 2t at the point (0, π, 1).
The equation of the normal plane is 4y = 4π, or equivalently, y = π.
What is osculating plane?The word osculate comes from the Latin osculatus, which is a past participle of the verb osculari, which means "to kiss." Thus, an osculating plane is one that "kisses" a submanifold.
To find the osculating plane and normal plane of the curve x = sin 2t, y = t, z = cos 2t at the point (0, π, 1), we need to follow these steps:
Find the first and second derivatives of the curve with respect to t.Evaluate the derivatives at t = π to get the velocity, acceleration, and curvature vectors at the point (0, π, 1).Use the velocity and acceleration vectors to find the normal vector of the osculating plane.Use the normal vector and the point (0, π, 1) to find the equation of the osculating plane.Use the curvature vector to find the normal vector of the normal plane.Use the normal vector and the point (0, π, 1) to find the equation of the normal plane.Step 1: Find the first and second derivatives of the curve with respect to t.
x' = 2cos2t
y' = 1
z' = -2sin2t
x'' = -4sin2t
y'' = 0
z'' = -4cos2t
Step 2: Evaluate the derivatives at t = π.
x'(π) = 2cos2π = 2
y'(π) = 1
z'(π) = -2sin2π = 0
x''(π) = -4sin2π = 0
y''(π) = 0
z''(π) = -4cos2π = -4
So the velocity vector at the point (0, π, 1) is v = ⟨2, 1, 0⟩, the acceleration vector is a = ⟨0, 0, -4⟩, and the curvature vector is κv = ⟨0, 4, 0⟩.
Step 3: Use the velocity and acceleration vectors to find the normal vector of the osculating plane.
The normal vector of the osculating plane is given by the cross product of the velocity and acceleration vectors:
n = v × a = ⟨2, 1, 0⟩ × ⟨0, 0, -4⟩ = ⟨4, 0, 0⟩
Step 4: Use the normal vector and the point (0, π, 1) to find the equation of the osculating plane.
The equation of the osculating plane is given by:
4(x - 0) + 0(y - π) + 0(z - 1) = 0
Simplifying, we get:
4x - 4 = 0
So the equation of the osculating plane is 4x = 4, or equivalently, x = 1.
Step 5: Use the curvature vector to find the normal vector of the normal plane.
The normal vector of the normal plane is given by the curvature vector:
n' = κv = ⟨0, 4, 0⟩
Step 6: Use the normal vector and the point (0, π, 1) to find the equation of the normal plane.
The equation of the normal plane is given by:
0(x - 0) + 4(y - π) + 0(z - 1) = 0
Simplifying, we get:
4y - 4π = 0
So, the equation of the normal plane is 4y = 4π, or equivalently, y = π.
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Regis has a bag with 8 tiles numbered
1 through 8. He randomly draws one tile
from the bag without looking Which of
the following describes a likely outcome?
A. He selects a tile with the number 0.
B. He selects a tile with the number 4.
C. He selects a tile with a number
greater than 7.
D. He selects a tile with a number
less than 6.
The outcome that Regis is likely to get after randomly drawing one tile from the bag would be 0. That is option A.
How to calculate the outcome of that event?To calculate the outcome of the event is to calculate the probability of selecting a tile with a number when one tile is drawn at random.
Probability = possible outcome/sample space.
Possible outcome = 1
sample space = 8
probability = 1/8 = 0.125
The probability is approximately = 0
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Brian cut out 15 paper shapes. Two thirds of the shapes were circles. The rest were triangles. How many shapes were triangles?
Answer: 5 were triangles :)
Find the gradients of lines A and B
The correct answer is (1,1) because both of the lines meet together at these numbers
Which ordered pairs lie on the graph of the exponential function f(x)=−32x+5
?
The ordered pair (0, 2) lies on the graph.
The ordered pair (-1, 0) lies on the graph.
The ordered pair (-1, 0) lies on the graph.
What is function?
In mathematics, a function is a relationship between two sets of elements, called the domain and the range, such that each element in the domain is associated with a unique element in the range.
The exponential function f(x) = -32x+5 can be written in the form of
f(x) = [tex]a(b^{x})+c[/tex], where a, b, and c are constants. Comparing with the given function, we have a = -3, b = 2, and c = 5.
To find the ordered pairs that lie on the graph of the exponential function, we can plug in different values of x and calculate the corresponding values of y using the function. Here are some examples:
When x = 0, we have f(0) = [tex]-3(2^{0})[/tex] + 5 = 2. Therefore, the ordered pair (0, 2) lies on the graph.
When x = 1, we have f(1) = [tex]-3(2^{1})[/tex] + 5 = -1. Therefore, the ordered pair (1, -1) lies on the graph.
When x = -1, we have f(-1) = [tex]-3(2^{-1})[/tex] + 5 = 6/2 - 3 = 0. Therefore, the ordered pair (-1, 0) lies on the graph.
We can continue this process to find more ordered pairs that lie on the graph of the exponential function.
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Need help please
The half-life of Palladium-100 is 4 days. After 16 days a sample of Palladium-100 has been reduced to a mass of 2 mg.
What was the initial mass (in mg) of the sample? --------------
What is the mass 7 weeks after the start?-------------
Suppose that the functions fand g are defined as follows.
f(x)=2x-1
g(x)=√3x-5
The composite functions (f/g)(x) and (f-g)(x) are (2x-1)/√(3x-5) and (2x-1) -√(3x-5)
Calculating the composite functions (f/g)(x) and (f-g)(x)To calculate (f/g)(x), we need to divide f(x) by g(x):
(f/g)(x) = f(x)/g(x) = (2x-1)/√(3x-5)
The domain of (f/g)(x) is the set of all x-values for which the denominator √(3x-5) is not equal to zero and non-negative
3x-5 ≥ 0, or x ≥ 5/3
Therefore, the domain of (f/g)(x) is x ≥ 5/3.
To calculate (f-g)(x), we need to subtract g(x) from f(x):
(f-g)(x) = f(x) - g(x) = (2x-1) - √(3x-5)
The domain of (f-g)(x) is the set of all x-values for which the expression inside the square root is non-negative:
3x-5 ≥ 0, or x ≥ 5/3
Therefore, the domain of (f-g)(x) is x ≥ 5/3.
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Find the area of the triangle. Plsss answer
Answer:
1. 90
2. 468
Step-by-step explanation:
Answer:
Question 1). 45 in
Question 2). 234 m
Step-by-step explanation:
Area = base x height x 1/2
10 x 9 = 90
90 x 1/2 = 45
Question 2)
30 x 15.6 = 468
468 x 1/2 = 234 m
Which statements describe a parallelogram that must be a rectangle?
Select each correct answer.
parallelogram with a pair of congruent consecutive sides
parallelogram with congruent diagonals
parallelogram with opposite sides congruent
• parallelogram with a right angle
• parallelogram with perpendicular diagonals
Answer:
A parallelogram with congruent diagonals, a parallelogram with a right angle, a parallelogram with perpendicular diagonals.
Step-by-step explanation:
By definition, a rectangle is nothing but a parallelogram with one right angle. One of the properties of a rectangle is that its perpendiculars are congruent. Finally, the diagonals of a rectangle are congruent (this can be proved by finding congruent triangles split by the diagonals, using CPCTC, etc.).
Determine if the series converges or diverges. If the series converges, find its sum.
80
Σ
5
n(n+3)
OA. The series converges to
B. The series converges to
55
18
25
6
Hence,the sum of this series is approximately 1.193.
What is the series?A series in mathematics is essentially the process of adding an unlimited number of quantities, one after the other, to a specified initial amount. A significant component of calculus and its generalization, mathematical analysis, is the study of series.
What is the convergence series?If a series partial sum sequence tends to a limit, it is said to be convergent (or to converge); this indicates that if one adds partial sums one after the other in the order indicated by the indices, they approach closer and closer to a specified number.
Comparing the two series will help us better understand how they differ.
[tex]\lim_{n \to \infty} \frac{\frac{5}{n(n+3)}}{\frac{1}{n^2}}= \lim_{n \to \infty} \frac{5n^2}{n(n+3)}= \lim_{n \to \infty} \frac{5n}{n+3}= 5[/tex]
Since both series either converge or diverge together, the limit is a positive finite number. The supplied series [tex]\sum_{n=1}^{\infty} \frac{1}{n^2}[/tex]also converges because the series [tex]\sum_{n=1}^{\infty} \frac{5}{n(n+3)}[/tex] converges (by the p-series test with p = 2).
We can employ the partial fraction decomposition to determine the sum:
[tex]\frac{5}{n(n+3)} = \frac{1}{n} - \frac{1}{n+3}[/tex]
Consequently, we have
[tex]\sum_{n=1}^{\infty} \frac{5}{n(n+3)} \\= \sum_{n=1}^{\infty} \left(\frac{1}{n} - \frac{1}{n+3}\right)\\= \left(1 - \frac{1}{4}\right) + \left(\frac{1}{2} - \frac{1}{5}\right) + \left(\frac{1}{3} - \frac{1}{6}\right) + \dots\\[/tex]
[tex]= \frac{3}{4}+ \frac{3}{10}+ \frac{1}{6} + \dots\\[/tex]
The harmonic series of corresponding terms and the sequence [tex]0, 0, \frac{1}{6}, 0, 0,\frac{1}{30}, 0, 0, \frac{1}{42}, 0, 0,\frac{1}{66},...[/tex] can be added to find the sum of this series. The total is approximately 1.193.
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A rectangle has an area of 104cm^2 and a width that measures 8 cm. Find the perimeter of the rectangle.
Answer:
p=42
Step-by-step explanation:
S=104 sm^2
a=8 sm
S=a*b
104 sm^2=8*b
b=104 sm^2/8 sm=13 sm
P=2*(a+b)=2*(13 sm+8 sm)=42 sm
What 2 numbers add up to 13 but multiply to -48??
Answer:
3 and -16
Step-by-step explanation:
To find two numbers that add up to 13 but multiply to -48, we can start by making a list of the factors of -48:
1, -1, 2, -2, 3, -3, 4, -4, 6, -6, 8, -8, 12, -12, 16, -16, 24, -24, 48, -48
We can see that the only two numbers in this list whose sum is 13 are 3 and -16. To verify that these numbers multiply to -48, we can simply multiply them together:
3 x (-16) = -48
Therefore, the two numbers that add up to 13 but multiply to -48 are 3 and -16.
Answer: -3, 16
Step-by-step explanation:
Calculate the volume of a sphere with a diameter of 4.
Answer: 33.51 cubic units.
Step-by-step explanation: The equation for the volume of a circle is:
V = (4/3)πr³
where r is the sweep of the circle.
Since the distance across of the circle is given as 4, we are able discover the span by partitioning the distance across by 2:
r = d/2 = 4/2 = 2
Presently we are able plug within the esteem of the sweep into the equation for the volume:
V = (4/3)π(2³) = (4/3)π(8) = 32/3π
Answer: 33.51
Step-by-step explanation:
The formula is 4/3 pi r^3. The radius is 2. If you do the equation, you get roughly 33.51.
Cual es el valor de p(B/A)?
The value of p(B/A) represents the probability that event B will occur given that event A has occurred.
What is the probability?It is a measure of conditional probability, which is calculated using the formula:
p(B/A) = p(A ∩ B) / p(A)
Where:
p(A ∩ B) is the probability that both events A and B occur together, that is, the intersection of A and B.p(A) is the probability that event A will occur.Therefore, To calculate the value of p(B/A), you need to know the probabilities of A and B, as well as the probability that both occur together.
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m 32 33 There are red tiles and blue tiles in a box. The ratio of red tiles to blue tiles is 3:5. There are 12 more blue tiles than red tiles in the box. How many red tiles are in the box? A 18 B C 20 30 D 48 What is the surface area, in square inches, of the rectangular prism formed by folding the net below? 8 in. 23 in. 8 in. 36 in.
The number of red tiles in the box given the chance ratio of red to blue tiles is 18. The surface area of the rectangular prism is 2600 square inches.
Number of red tiles = x
Number of blue tiles = 12 + x
Total tiles = x + 12 + x
= 12 + 2x
Ratio of red = 3
Ratio of blue = 5
Total ratio = 3 + 5 = 8
Number of red tiles = 3 / 8 × 12+2x
x = 3(12 + 2x) / 8
x = (36 + 6x) / 8
8x = 36 + 6x
8x - 6x = 36
2x = 36
x = 36/2
x = 18 tiles
Therefore, The number of red tiles in the box given the chance ratio of red to blue tiles is 18.
b) To find the surface area of the rectangular prism, we need to find the area of each of its faces and add them together. Looking at the net, we see that there are three pairs of identical rectangles: the top and bottom faces, the front and back faces, and the left and right faces. Each of these rectangles has dimensions of 23 inches by 8 inches.
Therefore, the surface area of the rectangular prism is:
=2 * (23 in. * 8 in.) (top and bottom faces)
=2 * (36 in. * 8 in.) (front and back faces)
=2 * (23 in. * 36 in.) (left and right faces)
= 368 + 576 + 1656
= 2600 square inches
Therefore, the surface area of the rectangular prism is 2600 square inches.
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In a recent survey, people were asked whether they would prefer to work flexible hour
- even when it meant slower career advancement-so they could spend more time with their
families. The figure shows the results of the survey. What is the probability that four people chosen at random would prefer flexible work hours? (Round your answer to four decimals)
The probability of the given situation through which the given relation is satisfied is 0.78
What about probability?Probability is a measure of the likelihood or chance of an event occurring. It is expressed as a number between 0 and 1, where 0 indicates that the event is impossible, and 1 indicates that the event is certain to occur.
The probability of an event can be calculated by dividing the number of favorable outcomes by the total number of possible outcomes. For example, if you flip a fair coin, there are two possible outcomes: heads or tails. The probability of getting heads is 1/2, or 0.5, since there is one favorable outcome (heads) out of two possible outcomes (heads or tails).
Probabilities can also be expressed as percentages or fractions. For example, a probability of 0.25 can be expressed as 25%, or as a fraction of 1/4.
Probability theory is a branch of mathematics that deals with the analysis of random events and the quantification of uncertainty. It has applications in a wide range of fields, including statistics, physics, finance, and engineering.
According to the given information:Flexible hour work = 78%
Don't Know = 9%
Rigid hour = 13%
The probability of that 4 people choose flexible hour for work is,
0.78
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I need help solving this thank you
The negation is the fourth option.
6 + 3 ≠ 9 or 6 - 3 ≠ 9
How to write the negation?The negation of an equation is an inequality such that we just change the equal sign, by the "≠" sign.
Here we start with the two equations.
6 + 3 = 9 or 6 - 3 = 9
Just change the equal signs for different signs:
6 + 3 ≠ 9 or 6 - 3 ≠ 9
That is the negation, fourth option.
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help please! state the key features for the graph
Answer:
Axis of symmetry =1
vertex =(1,2)
y intercept =0
min/max= -6,2
domain= 0,1,2
range =y≥1,2
find the area and perimeter of each figure below.
Answer:
finding the perimeter, you sumthe distance all round that is 7+7.5+17.8+6=38.3
38.3 is the perimeter
I need help
A population of bacteria is growing according to the equation p(t)=800e^0.14t Estimate when the population will exceed 1151.
t= ---------
Answer: t=2.59
Step-by-step explanation:
This is a matter of clearing out the equation
set 1151=800e^0.14t
1151/800=e^0.14t
ln(1151/800)/0.14=t
t=2.59
Write the polynomial function of least degree that has zeros of x=0, x= 2i and x =3
(assume all coefficients must be real)
A. x)=x²-3x³+4x² - 12x
B. x)=x²-3x² + 4x-12
C. x)=x²-3x³+4x² + 12x
D. f(x)=x² + 3x² - 6x + 12
The polynomial function of least degree that has zeros of x=0, x=2i, and x=3, and with all coefficients real is:
f(x) = x² - 3x³ + 4x² - 12xHow to find the polynomialSince the zeros of the polynomial function are given as
x=0, x=2i, and x=3,
we can write the function in factored form as follows:
f(x) = a(x-0)(x-2i)(x-3)
where
a is a constant coefficient and the factors correspond to the given zeros.
Since all coefficients must be real, we know that the complex conjugate of 2i, which is -2i, must also be a zero of the function. Therefore, we can rewrite the function as:
f(x) = a(x-0)(x-2i)(x+2i)(x-3)
Expanding this expression gives:
f(x) = a(x² + 4)(x-3)
Multiplying out the brackets and collecting like terms, we get:
f(x) = ax³ - 3ax² + 4ax - 12a
To find the value of 'a', we can use the fact that the coefficient of the x³ term is 1. Thus, we have:
a = 1/(1*4) = 1/4
Substituting this value of 'a' in the above expression, we get:
f(x) = (1/4)x³ - (3/4)x² + x - 3
Therefore, the polynomial function of least degree that has zeros of x=0, x=2i, and x=3, and with all coefficients real is:
Option A: f(x) = x² - 3x³ + 4x² - 12x
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Find the difference between the median and the mean by using 4.25, 6.25, 8,8,8, 4.25, 6,9,6, 8.25, 9.25
Check the picture below.
What is the area of the trapezoid?
Enter your answer in the box.
The area of given trapezoid whose parallel sides are 8 inches, 16 inches & its perpendicular height is 6 inches is 72 inches².
What is a trapezoid?
Quadrilaterals or four-sided polygons with one pair of parallel sides and one pair of non-parallel sides are known as trapezoids, usually spelt trapezium. It has a perimeter and covers a certain amount of space. The bases of the trapezium are the sides that are parallel to one another. Legs or lateral sides indicates to the non-parallel sides. The altitude or perpendicular height is the separation between the parallel sides. This shape's area is equal to half of the product of its parallel sides times its height.
Given dimensions of trapezoid:
let parallel sides be denoted by a , b & height be 'h'
a= 8 inches (top side)
b= 8+4+4=16 inches (bottom side)
h= 6 inches
Area = [tex]\frac{sum of parallel sides}{2}[/tex] x height
=[tex]\frac{(a+b)h}{2}[/tex]
=[tex]\frac{(8+16)6}{2}[/tex]
=[tex]\frac{24(6)}{2}[/tex]
=24 x 3
=72 sq. inches
The area of given trapezoid is 72 inches²
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Determine the interval(s) on which the function Is constant.
Write your answer as an interval or list of intervals.
When writing a list of Intervals, make sure to separate each interval with a comma and to use as few intervals as possible.
Click on "None* if applicable.
The intervals on which the function Is constant are [-4, -3] and [3, 6]
Determining the interval(s) on which the function Is constant.A function is considered constant over an interval if the function has the same output for all the inputs within that interval.
In other words, the function does not change over that interval.
Using the above as a guide, we have the following:
The function does not change over the intervals [-4, -3] and [3, 6]
Hence, the intervals are [-4, -3] and [3, 6]
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Please help me and ty :)
Answer:
$60
Step-by-step explanation:
In a set of data, the median is the value that is placed in the middle. to find the median, we must order the numbers from least to greatest, like so:
20, 40, 40, 50, 60, 60, 60, 60, 70, 70
Using this representation, we find that there are two numbers in the middle: 60 and 60 (since the amount of data is an even number, there will be two numbers in the center).
When something like this occurs, find the average of the two numbers in the middle (and the values and divide by the number of values). We have:
60 + 60 = 120;
[tex]\frac{120}{2} = 60[/tex]
Therefore, the median is $60.
Calculate the amount of simple interest earned. $6,000 at 12% for 7 years The interest is $
Answer:
$5040
Step-by-step explanation:
Apply the formula
SI = (Principal)(Rate)(Time)
= 6000×0.12×7
= $5,040