The probability of rolling two fair six-sided dice and getting a sum of 4 or higher is 11/12
To calculate the probability of rolling two fair six-sided dice and getting a sum of 4 or higher, we first need to calculate the total number of possible outcomes.
The number of possible outcomes when rolling two dice is 6 × 6 = 36, since each die has 6 possible outcomes.
Now, let's find the number of outcomes that result in a sum of 4 or higher. We can do this by listing all the possible outcomes:
Sum of 4: (1, 3), (2, 2), (3, 1) = 3 outcomes
Sum of 5: (1, 4), (2, 3), (3, 2), (4, 1) = 4 outcomes
Sum of 6: (1, 5), (2, 4), (3, 3), (4, 2), (5, 1) = 5 outcomes
Sum of 7: (1, 6), (2, 5), (3, 4), (4, 3), (5, 2), (6, 1) = 6 outcomes
Sum of 8: (2, 6), (3, 5), (4, 4), (5, 3), (6, 2) = 5 outcomes
Sum of 9: (3, 6), (4, 5), (5, 4), (6, 3) = 4 outcomes
Sum of 10: (4, 6), (5, 5), (6, 4) = 3 outcomes
Sum of 11: (5, 6), (6, 5) = 2 outcomes
Sum of 12: (6, 6) = 1 outcome
Therefore, the number of outcomes that result in a sum of 4 or higher is 3 + 4 + 5 + 6 + 5 + 4 + 3 + 2 + 1 = 33.
Therefore, the probability of rolling two fair six-sided dice and getting a sum of 4 or higher is 33/36 = 11/12.
To find the probability of getting a sum of 44 or higher, we need to subtract the probability of getting a sum of 43 or lower from 1:
Sum of 2: (1, 1) = 1 outcome
Sum of 3: (1, 2), (2, 1) = 2 outcomes
Sum of 4: (1, 3), (2, 2), (3, 1) = 3 outcomes
Sum of 5: (1, 4), (2, 3), (3, 2), (4, 1) = 4 outcomes
Sum of 6: (1, 5), (2, 4), (3, 3), (4, 2), (5, 1) = 5 outcomes
Sum of 7: (1, 6), (2, 5), (3, 4), (4, 3), (5, 2), (6, 1) = 6 outcomes
Sum of 8: (2, 6), (3, 5), (4, 4), (5, 3), (6, 2) = 5 outcomes
Sum of 9: (3, 6), (4, 5), (5, 4), (6, 3) = 4 outcomes
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Please please please answer this i need ittt
Answer: 81
Step-by-step explanation: this would be 81 because [tex]3^{3}[/tex] is basically 3 x 3 x 3 and 3 x 3 x 3=27 then multiply 27 by 3, and you get 81
Answer:
3^4. my answer needs to be 20+character sooooooooo
A fair 6-sides die is rolled 550 times. What is a reasonable prediction for the number of times the event of landing on an even number?
The prediction for the number of times the event of landing on an even number in 550 rolls is 275
Estimating the reasonable predictionFrom the question, we have the following parameters that can be used in our computation:
The number of times = 550
The sample space of a fair 6-sided die is
S = {1, 2, 3, 4, 5, 6}
And as such the even numbers are
Even = {2, 4, 6}
This means that in a fair 6-sided die, we have
P(Even) = 3/6
When evaluated, we have
P(Even) = 1/2
So, when the die is rolled 550 times, we have
Expected value = 1/2 * 550
Evaluate
Expected value = 275
Hence, the number of times is 275
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The Maclaurin series for a function f is given by f(x)=x−x^3/3!+x^5/5!−x^7/7!+⋯+(−1)^n*x^2n+1/(2n+1)!+⋯ and converges to f(x) for all x. Let g be the function defined by g(x)=f(x2)
The Maclaurin series for g(x) is given by g(x) =[tex]x^2 - x^6/3! + x^10/5! -[/tex] [tex]x^14/7![/tex] [tex]+ ⋯ + (-1)^n*x^(4n)/(2n+1)! + ⋯[/tex]
How to the Maclaurin series of g(x)?The function g(x) is defined as g(x) = [tex]f(x^2)[/tex], where f(x) is a function with a Maclaurin series expansion.
To find the Maclaurin series for g(x), we substitute [tex]x^2[/tex] into the Maclaurin series of f(x). The resulting series for g(x) is obtained by replacing each occurrence of x in the series for f(x) with x^2:
g(x) = [tex]f(x^2) = (x^2) - (x^2)^3/3! + (x^2)^5/5! - (x^2)^7/7! + ⋯ + (-1)^n*(x^2)^(2n+1)/(2n+1)! + ⋯[/tex]
Simplifying the terms, we have:
g(x) =[tex]x^2 - x^6/3! + x^10/5! - x^14/7! + ⋯ + (-1)^n*x^(4n+2)/(2n+1)! + ⋯[/tex]
This represents the Maclaurin series expansion for the function g(x) in terms of the original function f(x) with the argument squared.
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Quality-control research determined that of all new
cars sold by Sherman Motors, 8% will require a
minor repair during the first year of ownership.
Suppose you survey the owners of three cars from
Sherman Motors. Find the probability to the nearest
percent that exactly one car will require a minor
repair in the first year
The probability that out of three exactly one car will require a minor repair in the first year = 33%
P(E ) = no. of favorable outcome/total no. of outcome
E here represents exactly one car that requires a minor repair.
out of three cars, exactly one car will need a minor repair
No. of favorable outcome = 1
Total no. of outcome = 3
Now, putting value in P(E) we get
P(E) = 1/3
P(E) = 0.333
To get percentage we multiply by 100
P(E) = 0.333 × 100
P(E) = 33.3
The probability to the nearest percent = 33%
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Earthworm Rivals are building the set for
their new music video. There is a tower made
of 9 glowing bricks that stands 5. 4 meters tall. If each of the bricks is the same exact size,
how tall is each brick?
Since each of the bricks is the same exact size, then each brick is 0.6 meters tall.
To determine the height of each glowing brick, we need to divide the total height of the tower (5.4 meters) by the number of bricks (9). This gives us the average height of each brick.
Using the formula for division, we can write this as:
Height of each brick = Total height of tower / Number of bricks
Plugging in the given values, we get:
Height of each brick = 5.4 meters / 9 bricks
Simplifying this expression, we can cancel out the units of "bricks" to get:
Height of each brick = 0.6 meters
Therefore, each glowing brick in the tower is 0.6 meters tall.
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I need help please……..
A buyer purchases a 2022 Jeep Grand Wagoneer Series III for $109,000 and a 2023 Cadillac Escalade V for $149,000. How much money will he spend a year for car payments?
The amount of money that he will spend a year for car payment would be = $258,000
How to calculate the total amount of money that will be spent of car payments?To calculate the amount of money that will be used for car payment the following is carried out.
The cost for purchasing 2022 Jeep Grand Wagoneer Series III = $109,000
The cost for purchasing a 2023 Cadillac Escalade V = $149,000
The total amount spent of car payments = 109000+149000
= $258,000
Therefore, the buyer will spend up to $258,000 a year for car payment when the price for both cars he bought is being summed up.
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A square a rectangle have the same perimeter of a square has a side length of 8x units. The rectangle has a length of (5x + 12) and a width of 10 units. what will be the perimeter of both a rectangle and the square
Answer:
Step-by-step explanation:
The perimeter of a square is calculated by multiplying the length of one side by 4. Since the side length of the square is 8x units, the perimeter of the square is 4 * 8x = 32x units.
The perimeter of a rectangle is calculated by adding the lengths of all four sides or by using the formula 2 * (length + width). Since the length of the rectangle is (5x + 12) units and the width is 10 units, the perimeter of the rectangle is 2 * ((5x + 12) + 10) = 10x + 44 units.
Since both shapes have the same perimeter, we can set their perimeters equal to each other and solve for x:
32x = 10x + 44 22x = 44 x = 2
Substituting this value of x back into the expression for the perimeter of either shape, we find that the perimeter of both the square and the rectangle is 64 units.
Find f such that f(x) = 5/
. (16) = 49.
Let's find a function f(x) such that f(x) = 5x and f(16) = 49.
To find the function, we first plug in the given input (x = 16) and output (f(16) = 49):
49 = 5 * 16
Next, we solve for the unknown constant in the function:
49 = 80
5 = 49/80
Now, we have found the function f(x): f(x) = (49/80)x
The property that every input is related to exactly one output defines a function as a relationship between a set of inputs and a set of allowable outputs. A mapping from A to B will only be a function if every element in set A has one end and only one image in set B. Let A & B be any two non-empty sets.
Functions can also be defined as a relation "f" in which every element of set "A" is mapped to just one element of set "B." Additionally, there cannot be two pairs in a function that share the same first element.
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i need to figure it out
The solution to the equation 4x + 17 = 23 is x = 3/2.
How to solve the equationIt should be noted that to solve this equation, we need to isolate the variable x on one side of the equation.
First, we can subtract 17 from both sides of the equation:
4x + 17 - 17 = 23 - 17
Simplifying the left side of the equation:
4x = 6
Next, we can divide both sides of the equation by 4:
4x/4 = 6/4
Simplifying:
x = 3/2
Therefore, the solution to the equation 4x + 17 = 23 is x = 3/2.
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1) Last year a computer cost $1,600 to purchase. A year later the same computer cost $1,850 to purchase. By what percent did the cost of the computer increase? (please show your work)
2)A flock of 50 geese landed at a pond. Later that day there were 75 in the pond. By what percent did the number of geese increase?
a
80%
b
50%
c
2%
d
10%
Answer:
1) 15.625% increase. 1850-1600=250. 250/1600=0.15625. 0.15625x100= 15.625.
2) 50%
Step-by-step explanation:
1) 1850-1600=250. 250/1600=0.15625. 0.15625x100= 15.625.
2) 75-50=25. 25/50=0.5. 0.5x100=50
M
what is the rate of return when 30 shares of stock
a. purchased for $20/share, are sold for $720? the
commission on the sale is $6.
rate
return = [?] %
give your answer as a percent rounded to the
nearest tenth.
The rate of return is 19%, rounded to the nearest tenth.
Given that a purchased for $ 20/share, are sold for $ 720. $ 6 is the commission on the sale. We need to calculate the total cost of the investment and the total proceeds from the sale, and then use the formula for rate of return.
The total value should be
= 20 × 30
= $ 600
Since, it is sold for $ 720 along with commission of $6 so final money should be
= 720 - 6
= $ 714.
Now rate of return is
= (714 - 600)/714*100
= 114/600*100
= 19%
Therefore, the rate of return is 19%, rounded to the nearest tenth.
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Identify the line of symmetry for the function below:
g(x) = |x +9|- 11
Answer:
x = -9
Step-by-step explanation:
As this is an absolute value function, the line of symmetry is the x-value of the maximum/minimum point. An absolute value function can be denoted as y = |x - h| + k, where (h, k) is the maximum/minimum point. We only need the x-value of the maximum/minimum point, so we only have to look at "h". Now, we can use y = |x - h| + k and turn it into g(x):
y = |x - h| + k
g(x) = |x - -9| + -11 --> this means h = -9, and the line of symmetry is at x = -9
g(x) = |x + 9| - 11
Answer:
I think x equals --9
Find two vectors in opposite directions that are orthogonal to the vector u. (The answers are not unique. Enter your answer as a comma-separated list of vectors.) u = (5, -4,8) Determine whether the planes are orthogonal, parallel, or neither
The cross-product of u and v:
w = u × v = (5, -4, 8) × (-8, -4, 5) = (-20, -60, 32)
Thus, w is orthogonal to u. Since we need two vectors in opposite directions, we can negate w:
-w = (20, 60, -32)
Therefore, the two orthogonal vectors in opposite directions are w = (-20, -60, 32) and -w = (20, 60, -32).
To find two vectors that are orthogonal to u, we can use the cross-product. Let v = (4,5,0) and w = (-8,0,5). Then v x u = (40,40,45) and w x u = (20,-40,20). So two vectors orthogonal to u are (40,40,45) and (20,-40,20).
To determine whether two planes are orthogonal, parallel, or neither, we can look at the normal vectors of each plane. Let the first plane be defined by the equation 2x + 3y - z = 4 and the second plane being defined by the equation :
4x + 6y - 2z = 8.
The normal vector of the first plane is (2,3,-1) and the normal vector of the second plane is (4,6,-2).
Since the dot product of these two normal vectors is -2(3) + 3(6) - 1(2) = 14, which is not equal to 0, the planes are not orthogonal.
To determine if they are parallel, we can check if the ratio of their normal vectors is constant. Dividing the second normal vector by the first, we get (4/2, 6/3, -2/-1) = (2,2,2). Since this is a constant ratio, the planes are parallel.
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Jack starts to save at age 40 for a vacation home that he wants to buy for his 50th birthday. He will contribute $1000 each quarter to an account, which earns 2. 1% interest, compounded annually. What is the future value of this investment, rounded to the nearest dollar, when Jack is ready to purchase the vacation home?
$11,000
$11,231
$44,000
$44,924
The future value of the investment when Jack is ready to purchase the vacation home is $44,924.
To solve this problem, we can use the formula for future value of an annuity:
FV = Pmt x [(1 + r)^n - 1] / r
Where:
Pmt = $1000 (quarterly contribution)
r = 0.021 (annual interest rate)
n = 40 (number of quarters until Jack turns 50)
Plugging in the numbers, we get:
FV = $1000 x [(1 + 0.021)^40 - 1] / 0.021
FV = $44,924.38
Therefore, the future value of Jack's investment, rounded to the nearest dollar, is $44,924. So the correct answer is $44,924.
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Marsha is considering purchasing 3 points on a $350,000 home mortgage for 20 years. If she
purchases the 3 points, at a cost of 1 percent per point, her monthly mortgage would be
approximately $1,878.63. If she decides not to purchase any points, Mercedes' monthly
payment would be approximately $1,987.13. How much money will Mercedes save over the life
of the loan if she purchases the 3 points?
Marsha would save $26,040 over the life of the loan if she purchases the 3 points.
First, let's calculate the monthly payment if Marsha doesn't purchase any points. We can use a mortgage calculator or the PMT function in Excel to find;
PMT = $1,987.13
Now, let's calculate the monthly payment if Marsha purchases 3 points;
Loan amount = $350,000
Points cost = 3 points × 1% × $350,000 = $10,500
Effective loan amount = $350,000 - $10,500 = $339,500
Interest rate = 4.5% / 12 = 0.375%
Number of payments=20 years × 12 = 240
Using the PMT function, we get;
PMT = $1,878.63
So, by purchasing 3 points, Marsha can save;
$1,987.13 - $1,878.63 = $108.50 per month
Over the life of the loan, which is 20 years or 240 months, the total savings would be;
$108.50 × 240 = $26,040
Therefore, Marsha would save $26,040 amount of money.
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Can someone help me please
Answer:
A, 48 degrees
Step-by-step explanation:
180 - 42 - 90 = 48 (all interior angles add up to 48)
Answer:
it’s A
Step-by-step explanation:
it’s a because the right triangle is always 90°.As you can see there is 42° so you subtract to find the missing number.
A rectangular pyramid has a volume of 480 in. If a rectangular prism has a base and height congruent to the pyramid, what
is the volume of the prism? († point)
Please help!!!
After considering all the given data we come to the conclusion that the volume of the prism is 1440 in³, under the condition that A rectangular pyramid has a volume of 480 in.
The volume of a rectangular pyramid is represented by the formula
(1/3) × base area × height.
Now, the volume of a rectangular prism is given by the formula
base area × height.
Now if we consider the rectangular prism has a base and height congruent to the pyramid, then the base area of the prism is equivalent to the base area of the pyramid. Then, the volume of the prism is equivalent to three times that of the pyramid.
Hence, the volume of the pyramid is 480 in³, we can evaluate the volume of the prism
Volume of prism = 3 × Volume of pyramid
= 3 × (1/3) × Base area × Height
= Base area × Height
Then, the volume of the rectangular prism is
480 × 3
= 1440 in³.
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Identify the equation of the line that passes through the pair of points (0,4) and (6, −3) in slope-intercept form.
An equation of the line that passes through the pair of points (0,4) and (6, −3) in slope-intercept form is y = -7x/6 + 4.
How to determine an equation of this line?In Mathematics and Geometry, the point-slope form of a straight line can be calculated by using the following mathematical expression:
y - y₁ = m(x - x₁)
Where:
x and y represent the data points.m represent the slope.First of all, we would determine the slope of this line;
Slope (m) = (y₂ - y₁)/(x₂ - x₁)
Slope (m) = (-3 - 4)/(6 - 0)
Slope (m) = -7/6
At data point (0, 4) and a slope of -7/6, a linear equation for this line can be calculated by using the point-slope form as follows:
y - y₁ = m(x - x₁)
y - 4 = -7/6(x - 0)
y = -7x/6 + 4
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Let G be the center of the equilateral triangle XYZ. A dilation centered at G with scale factor -3/4 is applied to triangle XYZ, to obtain triangle X'Y'Z'. Let A be the area of the region that is contained in both triangles XYZ and X'Y'Z'. Find A/the area of XYZ.
The calculated value of the expression A/the area of XYZ is [tex]\frac{49\sqrt3}{216}[/tex]
Finding the value of A/the area of XYZFrom the question, we have the following parameters that can be used in our computation:
Center of the equilateral triangle XYZ = GDilation centered at G with scale factor = 3/4By the ratio of corresponding sides (see attachment for figure), we have
(x + 2y)/(2x + y) = 3/4
By comparison, we have
x + 2y = 3
2x + y = 4
This gives
(x, y) = (5/3, 2/3)
The triangles are equilateral triangles
So, we have
Area of XYZ = 1/2 * side length² * sin(60)
This gives
Area of XYZ = 1/2 * (2x + y)² * sin(60)
Substitute the known values in the above equation
Area of XYZ = 1/2 * (4)² * sin(60)
Evaluate
Area of XYZ = 4√3
The region A is a trapezoid
So, the area is
A = 1/2 * Sum of parallel sides * height
So, we have
A = 1/2 * (x + y) * (x² - y²)
Recall that (x, y) = (5/3, 2/3)
So, we have
A = 1/2 * (5/3 + 2/3) * ((5/3)² - (2/3)²)
Evaluate
A = 49/18
Finding A/the area of XYZ, we have
A/the area of XYZ = 49/18 ÷ 4√3
This gives
A/the area of XYZ = 49/72 ÷ √3
Rationalize
A/the area of XYZ = [tex]\frac{49\sqrt3}{216}[/tex]
Hence, the value of the expression is [tex]\frac{49\sqrt3}{216}[/tex]
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Complete question
Let G be the center of the equilateral triangle XYZ. A dilation centered at G with scale factor -3/4 is applied to triangle XYZ, to obtain triangle X'Y'Z'. Let A be the area of the region that is contained in both triangles XYZ and X'Y'Z'. Find A/the area of XYZ.
XY = 2x + y
X'Z' = x + 2y
Region A is a trapezoid with parallel sides y & x and height x² - y²
Sam has a cylindrical storage container 7 inches tall with a radius of 5 inches.
How much cat litter will fit in the container?
Use 3. 14 for π. Round your answer to the nearest tenth.
Answer: _____________ cubic inches
The cylindrical storage container can hold 549.5 cubic inches of cat litter.
To find the volume of the cylindrical storage container, we need to use the formula:
V = πr²h
Where:
π is a constant approximately equal to 3.14
r is the radius of the container
h is the height of the container
Substituting the given values, we get:
V = 3.14 x 5² x 7
V = 549.5 cubic inches
Since we know that the vessel can hold549.5 boxy elevation of cat waste, we can use this value to determine how important cat waste can fit in the vessel in other units. For illustration, if we want to know how numerous liters of cat waste can fit in the vessel, we can convert boxy elevation to liters using the conversion factor 1 liter = 61.02 boxy elevation. boxy elevation61.02 boxy elevation per liter = 8.998 liters thus, the spherical storehouse vessel can hold roughly 9 liters of cat waste.
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. given that z is a standard normal random variable, a positive value of z indicates that: question 2 options: a) the standard deviation of z is negative b) the probability associated with z is negative c) the value z is to the left of the mean d) the area between zero and z is negative. e) the value z is to the right of the mean
The positive value of z indicates option e) the value z is to the right of the mean.
A standard normal random variable has a mean of 0 and a standard deviation of 1. Positive values of z represent values above the mean, while negative values of z represent values below the mean.
The probability associated with a value of z is always positive since it represents the likelihood of observing a certain value. The area between zero and z is also always positive since it represents the probability of observing a value between 0 and z.
Therefore, option e) is the correct answer as it reflects the relationship between positive values of z and their location relative to the mean.
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Translate the following statement into a mathematical equation:
Five times a number, minus three, is twelve.
Its translation is 5×3-3=12
f(x)=-x^(2)-8x+19
1.whats the functions minimum value?
2.where does the minimum value occur?
The minimum value of the function is -13 and the minimum value of the function occurs at the point (4, -13).
The function F(x) is a quadratic function with a negative coefficient of the squared term.
Therefore, the function has a maximum value.
To find the maximum value, we need to find the vertex of the parabola.
The x-coordinate of the vertex is given by x = -b/2a, where a and b are the coefficients of the x² and x terms respectively.
In this case, a = -1 and b = -8, so x = -(-8)/(2(-1)) = 4.
To find the minimum value, we substitute this x-value into the function to get F(4) = -(4²) - 8(4) + 19 = -13.
Therefore, the minimum value of the function is -13.
We found in part (1) that the x-coordinate of the vertex is x = 4.
To find the y-coordinate, we substitute this x-value into the function to get F(4) = -13.
Therefore, the minimum value of the function occurs at the point (4, -13).
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from the following quadratic function , g(x)= -4(x+2)^2-3 identify the difference between its parent function f(x)=x^2
Thus, through the steps of horizontal translation, dilation and at last vertical translation, the new quadratic function , g(x)= -4(x+2)²-3 from the parent function f(x)=x².
Explain about the parent function:The simplest function which nonetheless complies with a particular type of function's definition is a parent function. For instance, y = x would be the parent function when considering the linear functions that make a family of functions. The most basic linear function is this one.
In addition, by applying various transformations to the graph of the parent function, all of the functions in a family of functions can also be derived from it. Vertical shifts, extending or compressing both horizontally and vertically, reflecting and over x or y axes, and horizontal shifts are some of these transformations.Given parent function: f(x)=x²
new quadratic function , g(x)= -4(x+2)²-3
there is the translation of 2 units to right such that 2 is added to x.Now, there is dilation with the scale factor of -4.At last the function is shifted 3 units downThus, through the steps of horizontal translation, dilation and at last vertical translation, the new quadratic function , g(x)= -4(x+2)²-3 from the parent function f(x)=x².
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Complete question:
from the following quadratic function , g(x)= -4(x+2)²-3 .identify the difference between its parent function f(x)=x² and g(x).
PLEASE HELP AND SHOW WORK!! 10 PTS IF U ANSWER
Answer:
Step-by-step explanation:
You're going to want to break up the shape into three parts, two triangles, and the rectangle.
Starting with the left-most triangle: A=(L*W)/2
The length is 4ft and the width is 3ft, multiply and divide by 2 to get: A=6 square feet.
Do the same with the second triangle on the bottom left (L=2ft, W=2ft) to get A=2 square feet.
Now the rectangle, A=L*W and total length is 10ft (8ft+2ft) and the width is 3ft. Multiply these values to get A=30 square feet.
Last step: add up all three areas for the total area of the entire shape, 6+2+30=38.
Area= 38 square feet.
Which fraction is equivalent to a whole number select all that apply? 9/3, -16/8, 7/0, -5/3, 0/5
The fraction is equivalent to a whole number are 9/3, -16/8, 7/0, 0/5
What is a fraction?A fraction can simply be described as the part of a whole variable, a whole numbers, or a whole element.
In mathematics, there are different types of fractions. These fractions are listed thus;
Simple fractionsProper fractionsImproper fractionsComplex fractionsMixed fractionsFrom the information given, we have that;
Equivalent expressions or fractions are fractions with the same solutions
Then, we have;
9/3
Divide the values
3
-16/8
-2
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Mr. Lee has a small apple orchard. There are 7 rows of tree with n trees in each row. which two expression show different ways to find the total number of trees in Mr. Lee apple orchard?
Therefore , the solution of the given problem of expressions comes out to be 7n.
What exactly is an expression?Instead of using random estimates, it is preferable to use shifting numbers that may also prove increasing, reducing, variable or blocking. They could only help one another by trading tools, information, or solutions to issues. The justifications, components, or quantitative comments for tactics like further disagreement, production, and blending may be included in the assertion of truth equation.
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By dividing the number of rows by the number of trees in each row, one can calculate the total number of trees in Mr. Lee's apple orchard. Here are two expressions that demonstrate various approaches to determining the overall number of trees:
There are 7n = trees in all.
=> Total number of trees = (Number of rows) x (Number of trees in each row) = 7n
The total number of trees in the orchard is the outcome of both expressions.
While the second statement more directly depicts the multiplication, the first expression merely merges the two elements into a single term.
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FILL IN THE BLANK. Use part I of the Fundamental Theorem of Calculus to find the derivative of f(x) = x∫4 1/1+4t⁴ dt f'(x)=________
The derivative of f(x) is: f'(x) = [tan⁻¹(2)/2] - [tan⁻¹(1/2)/2]
The Fundamental Theorem of Calculus is a pair of theorems that link the concept of differentiation and integration. It states that if a function f(x) is continuous on an interval [a, b] and F(x) is the antiderivative of f(x) on the same interval, then:
Part I: The derivative of the integral of f(x) from a to x is equal to f(x):
d/dx ∫a to x[tex]f(t) dt = f(x)[/tex]
Part II: The integral of the derivative of a function f(x) on an interval [a, b] is equal to the difference between the values of the function at the endpoints of the interval:
∫a to b [tex]f'(x) dx = f(b) - f(a)[/tex]
Using Part I of the Fundamental Theorem of Calculus, we have:
f(x) = x∫4 1/(1+4t⁴) dt
Then, by the Chain Rule, we have:
f'(x) = d/dx [x∫4 1/(1+4t⁴) dt] = ∫4 d/dx [x(1/(1+4t⁴))] dt
= ∫4 (1/(1+4t⁴)) dt
= [tan⁻¹(2t)/2]₄¹
= [tan⁻¹(2)/2] - [tan⁻¹(1/2)/2]
Therefore, the derivative of f(x) is:
f'(x) = [tan⁻¹(2)/2] - [tan⁻¹(1/2)/2]
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The sequence 10, 9. 5, 9. 0, 8. 5,. Has a common difference on
The sequence 200, 100, 50, 25,
has a common ratio of
The sequence 10, 9. 5, 9. 0, 8. 5,. has a common difference on -0.5
The sequence 200, 100, 50, 25, has a common ratio of 1/2
Let's start by discussing the sequence 10, 9.5, 9.0, 8.5. We can observe that each term is decreasing by 0.5. This means that the sequence has a common difference of -0.5.
In mathematical terms, the common difference is the constant value that is added or subtracted from each term in the sequence to obtain the next term. In this case, we can write the sequence as:
10, 10 - 0.5, 10 - 1.0, 10 - 1.5
where the common difference is -0.5.
Now, let's consider the sequence 200, 100, 50, 25. We can observe that each term is obtained by dividing the previous term by 2. This means that the sequence has a common ratio of 1/2.
In mathematical terms, the common ratio is the constant value that is multiplied by each term in the sequence to obtain the next term. In this case, we can write the sequence as:
200, 200/2, (200/2)/2, ((200/2)/2)/2
where the common ratio is 1/2.
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