The estimate for f(3.1, -3.01) using the linear approximation is approximately -54.28.
a. To find the linear approximation of f(x,y) at the point (3,-3), we need to find the gradient of the function at that point.
∇f(x,y) = [-8x, 4y]
So, at the point (3,-3), the gradient is ∇f(3,-3) = [-24, -12].
The linear approximation is given by:
L(x,y) = f(3,-3) + ∇f(3,-3)·(x-3,y+3)
Plugging in the values, we get:
L(x,y) = -4(3)^2 + 2(-3)^2 - 24(x-3) - 12(y+3)
Simplifying, we get:
L(x,y) = -12x - 4y - 36
b. To estimate f(3.1, -3.01) using the linear approximation, we plug in the values into the equation we found in part (a):
L(3.1, -3.01) = -12(3.1) - 4(-3.01) - 36
L(3.1, -3.01) ≈ -54.28
Therefore, the estimate for f(3.1, -3.01) using the linear approximation is approximately -54.28.
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A right rectangular prism has a length of 1 foot, a width of 1 3/8 feet, and a height of 5/8 foot. unit cubes with side lengths of 1/8 foot are added to completely fill the prism with no space remaining. how many cubes can fit inside the prism? explain how to find the number by using the volume formula. explain how to find the number by using the side lengths of the prism and the cubes.
The number of cubes of dimensions of [tex]\frac{1}{8}[/tex] foot that can fit into a rectangular prism of a length of 1 foot, a width of 1 [tex]\frac{3}{8}[/tex] foot, and a height of [tex]\frac{5}{8}[/tex] foot is 55.
Volume of cuboid = l * b * h
where l is the length
b is the breadth
h is the height
l = 1 foot
b = 1 [tex]\frac{3}{8}[/tex] foot = [tex]\frac{11}{8}[/tex] foot
h = [tex]\frac{5}{8}[/tex] foot
Volume of cuboid = 1 * [tex]\frac{11}{8}[/tex] * [tex]\frac{5}{8}[/tex]
= [tex]\frac{55}{64}[/tex] cubic foot
Volume of cube = [tex]s^3[/tex]
where s is the side
s = [tex]\frac{1}{8}[/tex] foot
Volume = [tex]\frac{1}{8}^3[/tex]
= [tex]\frac{1}{64}[/tex] cubic foot
Number of cubes = [tex]\frac{\frac{55}{64} }{\frac{1}{64} }[/tex]
= 55
The number of cubes that fit into the prism is 55.
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If the smith children are served randomly how likely is it that the two oldest smith children are served before the others ?
The probability of the two oldest Smith children being served first is 2/4 or 50%.
How likely is it for the two oldest Smith children to be served first?Assuming that all the children have an equal chance of being served first, there are a total of 4 possibilities for the order in which the two oldest Smith children can be served:
Oldest served first, followed by second oldestSecond oldest served first, followed by oldestOldest served first, followed by one of the younger children, then second oldestSecond oldest served first, followed by one of the younger children, then oldestOut of these 4 possibilities, only the first 2 would satisfy the condition that the two oldest Smith children are served before the others. So the probability that the two oldest Smith children are served first is 2/4 or 50%.
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State the null and alternative hypotheses you would use to test the following situation. The average time it takes for a person to experience pain relief from a certain pain reliever is 15 minutes. A new ingredient is added to help speed up pain relief and an experiment is conducted to test whether the new product does indeed speed up pain relief. What are the appropriate null and alternative hypotheses for the experiment
The appropriate null and alternative hypotheses for the experiment are:
Null hypothesis (H0): According to null hypothesis, the average time it takes for a person to experience pain relief from the new pain reliever is not beyond 15 minutes.
Alternative hypothesis (Ha): According to Alternative hypothesis, the average time it takes for a person to experience pain relief from the new pain reliever is significantly less than 15 minutes, indicating that the new ingredient does speed up pain relief.
The alternative and null hypotheses can be written as follows in symbols:
H0: = 15 (where μ is the population mean time for pain relief from the new pain reliever)
Ha: μ < 15
The one-tailed hypothesis test assumes that the new component of the painkiller can only reduce the duration of pain alleviation, not lengthen it. As a result, the rejection region will be in the left tail of the distribution, while the alternative hypothesis is one-tailed to the left.
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6.at what interval does the car reach the the highest acceleration ?
7. what is the highest acceleration of car ?
8. what is the lowest acceleration of the car ?
9. at what time interval did the car attains the lowest acceleration ?
10. base on the given data , how do u describe the motion of the car in the whole trip?
pls answer this thanks
I'm sorry, but you have not provided any data or information about the car's motion. Without this information, I cannot answer your questions accurately. Please provide more details or context about the car's motion.
I am unable to answer these specific questions without any given data. Please provide the data related to the car's acceleration, and I will be happy to help you with the analysis.
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Which system of equations is represented by the graph?
Answer:
Absoblute value Reflection
Step-by-step explanation:
You can tell it's absolbute value because of the parbolas, and it's reflected acroos the two points. Give brainliest please!
3. the sketch shows three adjacent angles. please write an equation that can be
used to determine the value of y?
28
36
please help
The value of y is 116°.
Three given Adjacent angles are 28°, 36° and y°.
When two angles are adjacent, they have a common side between them and share a common vertex where the two sides meet. The angles are said to be adjacent because they are next to each other.
We know that if a ray stands on a straight line, then the sum of adjacent angles is 180°.
So, 28° + y° + 36° = 180°
y° + 64° = 180°
Subtracting 64° from both sides.
y°- 64° = 180° - 64°
y° = 116°
Hence, the value of y is 116°.
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find f(x) for the given function.
f(x) = 1/1-x
To find f(x) for the given function f(x) = 1/1-x, we simply replace the "x" in the equation with whatever input value we want to evaluate the function at.
For example, if we want to find f(2), we would replace x with 2 and get:
f(2) = 1/1-2
f(2) = -1
Similarly, if we want to find f(a), we would replace x with a and get:
f(a) = 1/1-a
Therefore, f(x) = 1/1-x, where x is any input value.
To find f(x) for the given function, simply rewrite the function as it is:
f(x) = 1 / (1 - x)
Here, f(x) represents the function value for any input x, and the expression on the right side indicates how to calculate it.
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Put these numbers in order, from least to greatest. If you get stuck, consider using the number line.
3. 5 -1 4. 8 -1. 5 -0. 5 4. 2 0. 5 -2. 1 -3. 5
Write two numbers that are opposites and each more than 6 units away from 0
To put the numbers in order from least to greatest, we can use the number line: -3.5 -2.1 -1 -0.5 0.5 2 4 4.2 5 5.8 Two numbers that are opposites and each more than 6 units away from 0 are -7 and 7.
First, let's put the numbers in order from least to greatest:
-3.5, -2.1, -1.5, -1, -0.5, 0.5, 3.5, 4, 4.2, 4.8, 5
Now, let's find two numbers that are opposites and each more than 6 units away from 0. One example would be -7 and 7. These numbers are opposites (since they have the same magnitude but different signs), and they are both more than 6 units away from 0.
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Cameron and Lindsey need to make 160 cookies for the bake sale. Cameron made 2/5 of the cookies Lindsey made 16 cookies What fraction of the cookies do they still need to make?
The fraction of the cookies do they still need to made after Cameron and Lindsey made 80 cookies is 1/2.
Fractions are referred to as the components of a whole in mathematics. A single thing or a collection of objects might be the entire. When we cut a slice of cake in real life from the entire cake, the part represents the percent of the cake. The word "fraction" is derived from Roman. "Fractus" means "broken" in Latin.
The fraction was expressed verbally in earlier times. It was afterwards presented in numerical form. A piece or sector of any quantity is another name for the fraction.
Total cookies to be made is 160
Out of which Cameron made 2/5 of the cookies so,
2/5 x 160 = 64 cookies are made by Cameron
Lindsey made 16 cookies
So total cookies that still need to be made is,
= 160 - (64 + 16) = 80
The fraction of cookies still need to made is 80/160 = 1/2 .
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Leland wants to paint a portrait for his parents using a photograph of them that is 5 inches wide by 8 inches high. He wants the portrait to be proportional to the photograph and 42 inches high. Which proportion can Leland use to find , the width he needs to use to make the portrait proportional?
The ratio to make the portrait proportional is width/42 = 5/8
Which proportion can Leland use to find , the width he needs to useFrom the question, we have the following parameters that can be used in our computation:
Photograph that is 5 inches wide by 8 inches high.
This means that
Ratio = width/height
Substitute the known values in the above equation, so, we have the following representation
Ratio = 5/8
Given that he wants the portrait to be proportional to the photograph and 42 inches high.
Then, we have
width/42 = 5/8
Hence, the ratio to make the portrait proportional is width/42 = 5/8
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1. Liz Reynolds deposited $2,000 into a savings account that pays 8 % compounded quarterly. Complete the table to compute the amount in the account after 1 year.
The amount at end of the second quarter = [tex]42,448$ +2122,416$ = 2164.86$[/tex]
How to solveInterest for first quarter = [tex]2000 * 0,08*- = 40$\n4[/tex]
Amount at end of first quarter =[tex]40$ + 2000$ = 2040$[/tex]
Interest for second quarter = [tex]2040 * 0,08\n1\n* = 40,8$\n4[/tex]
Amount at end of second quarter = [tex]40,8$ + 2040$ = 2080,8$[/tex]
Interest for third quarter = [tex]2080,8 * 0,08*\n4\n= 41,616$[/tex]
Amount at end of third quarter = [tex]41,616$ +2080,0$ = 2122,416$[/tex]
1\nInterest for second quarter = [tex]2122,416 * 0,08 *-\n4\n= 42,44831$[/tex]
Amount at end of second quarter = [tex]42,448$ +2122,416$ = 2164.86$[/tex]
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Cardine company acquired and placed into use equipment on 2009 january 2, at a cash cost of 935,000. transportation charges amounted to 7,500, and installation and testing costs totaled 55,000. the equipment was estimated to have a useful life of nine years and a salvage value of 37,500 at the end of its life. it was further estimated that the equipment would be used in the production of 1,920,000 units of product during its life. during 2009, 426,000 units of product were produced. compute the depreciation if the year ended december 31, using straight line method.
t\The depreciation expense for the year ended December 31, 2009 using the straight-line method is $23,333.33.
The total cost of the equipment is $997,500 ($935,000 + $7,500 + $55,000). The depreciable cost is calculated by subtracting the salvage value from the total cost, which is $960,000 ($997,500 - $37,500).
To calculate the annual depreciation expense using the straight-line method, divide the depreciable cost by the useful life of the equipment. The annual depreciation expense is $106,666.67 ($960,000 ÷ 9).
Since the equipment was only used for a portion of the year, we need to prorate the annual depreciation expense based on the number of units produced.
The depreciation expense for the year ended December 31, 2009 is calculated as follows:
Depreciation Expense = (Annual Depreciation Expense ÷ Total Units of Production) x Units Produced in 2009
Depreciation Expense = ($106,666.67 ÷ 1,920,000) x 426,000
Depreciation Expense = $23,333.33
Therefore, the depreciation expense for the year ended December 31, 2009 using the straight-line method is $23,333.33.
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Find the area of this triangle.
round to the nearest tenth.
7 in
133
13 in
[ ? ) in2
Area of triangle = 45.5 in square ≈ 45.5 (rounded to the nearest tenth).
How to find area of triangle?The area of a triangle given its base and height.
The formula for the area of a triangle is:
Area = (1/2) x base x height
In this formula, "base" refers to the length of the side of the triangle that is perpendicular to the height, and "height" refers to the length of the line segment that is perpendicular to the base and passes through the opposite vertex.
In the problem you provided, the base of the triangle is given as 13 inches, and the height is given as 7 inches. So we can plug these values into the formula:
Area = (1/2) x 13 in x 7 in
= 45.5 in square
The units for area are square units, so we write the answer as "inches squared" or "in square".
Finally, we are asked to round the answer to the nearest tenth. Since there is only one decimal place in the answer, the "tenth" place is the same as the "one" place. Therefore, we look at the digit in the "one" place (which is 5) and round up to the nearest whole number. This gives us:
Area of triangle ≈ 45.5 (rounded to the nearest tenth).
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If the point P(-1,0) is a point on the terminal side of angle 0, an angle in standard position, what is the value of csc 0 ?
We know that the value of CSC 0 is undefined if the point P(-1,0) is a point on the terminal side of angle 0, an angle in standard position.
If the point P(-1,0) is a point on the terminal side of angle 0, an angle in standard position, then we can use the Pythagorean theorem to find the value of the hypotenuse. The hypotenuse (r) is the distance from the origin to point P, and can be found using the formula r = sqrt(x^2 + y^2), where x = -1 and y = 0.
r = sqrt((-1)^2 + 0^2) = 1
Since csc 0 is the reciprocal of sin 0, and sin 0 = 0/1 = 0, we have:
CSC 0 = 1/sin 0 = 1/0 = undefined.
Therefore, the value of CSC 0 is undefined if the point P(-1,0) is a point on the terminal side of angle 0, an angle in standard position.
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Si 12kg de papas y seis kg de arroz cuentan 102. 00 mientras que 9 kg de papas y 13 kilogramos de arroz cuestan 153. 00¿cuanto cuesta cada producto?
Entonces, concluimos que las papas cuestan 4 pesos por kilogramo y el arroz cuesta 9 pesos por kilogramo.
Para resolver este problema, necesitamos plantear un sistema de ecuaciones lineales con dos incógnitas: el precio por kilogramo de papas y el precio por kilogramo de arroz. Podemos escribir:
12p + 6r = 102
9p + 13r = 153
Donde p es el precio por kilogramo de papas y r es el precio por kilogramo de arroz. Podemos resolver este sistema de ecuaciones usando el método de eliminación. Multiplicando la primera ecuación por 13 y la segunda ecuación por -6, obtenemos:
156p + 78r = 1326
-54p - 78r = -918
Sumando estas ecuaciones, obtenemos: 102p = 408
Dividiendo ambos lados por 102, encontramos que p = 4. Por lo tanto, el precio por kilogramo de papas es de 4 pesos.
Para encontrar el precio por kilogramo de arroz, podemos sustituir p = 4 en una de las ecuaciones originales y resolver para r. Por ejemplo, usando la primera ecuación:
12(4) + 6r = 102
48 + 6r = 102
6r = 54
r = 9
Por lo tanto, el precio por kilogramo de arroz es de 9 pesos.
Entonces, concluimos que las papas cuestan 4 pesos por kilogramo y el arroz cuesta 9 pesos por kilogramo.
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What is the height, to the nearest tenth of a foot, of a tree that creates a 35-foot shadow when the sun is at an angle of elevation of 35⁰? (do NOT type in units, just the value)
The height of the tree is approximately 20.1 feet.
What is the height of a tree that produces a 35-foot shadow when the sun is at an angle of 35 degrees?To determine the height of the tree, we can use the tangent function, which relates the opposite side (the height of the tree) to the adjacent side (the length of the shadow) of a right triangle.
tan(35°) = height of tree / 35 feet shadow
Rearranging this formula, we get:
height of tree = 35 feet shadow x tan(35°)
Plugging in the given values and using a calculator, we get:
height of tree = 35 x tan(35°) ≈ 20.1 feet
Therefore, the height of the tree is approximately 20.1 feet.
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En una imprenta, 4 impresoras tardan 3 horas en imprimir 5000 hojas, ¿cuánto tiempo tardarán en imprimir 6000? escribe el resultado en horas con decimales
Tardarán aproximadamente 3.6 horas.
How long to print 6000 sheets?Para resolver este problema, podemos establecer una relación proporcional entre el número de hojas impresas y el tiempo requerido. Si 4 impresoras tardan 3 horas en imprimir 5000 hojas, podemos establecer la proporción
4 impresoras / 3 horas = 5000 hojas / x horas
Donde x representa el tiempo que tardarán en imprimir 6000 hojas. Podemos resolver esta proporción utilizando regla de tres:
4 / 3 = 5000 / x
Multiplicando en cruz, obtenemos:
4x = 3 * 5000
4x = 15000
x = 15000 / 4
x = 3750
Por lo tanto, tardarán aproximadamente 3750 horas en imprimir 6000 hojas.
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The owner of a company asks a manager, "How many hours does an employee work each week?" a. Is this a statistical question? Explain. The question a statistical question because there. Question 2 b. The manager records the number of hours worked each week by several employees. Display the data in a dot plot. Identify any clusters, peaks, or gaps in the data. Hours worked 40 39 40 39 47 38 41 37 40 41 40 43 40 41 40 38 40 42 40 42 Most of the data are clustered around. There is a peak at and a gap between. Question 3 c. Use the distribution of the data to answer the question. Most employees work about hours each week
This is a statistical question because it involves collecting data. Most employees work around 40 hours each week.
a. This is a statistical question because it involves collecting data on the number of hours worked by employees each week and analyzing the distribution or pattern of the data.
b. Here's a dot plot of the data provided:
37: •
38: ••
39: ••
40: ••••••••
41: •••
42: ••
43: •
47: •
The distribution of the data in the dot plot shows that the majority of employees worked around 40 hours each week, as this is where the data is most heavily clustered. Most of the data are clustered around 40 hours, with a peak at 40 hours. There is a gap between 43 and 47 hours.
c. Based on the distribution of the data, most employees work about 40 hours each week.
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Jack is making $9.50 per hour at his job re-stocking grocery shelves. His supervisor is impressed with his good work and gives him a 10% raise. How much per hour is Jack making now?
PLS HELP I WILL MARK YOU BRAINLIEST
Answer:
Jack now makes $10.45
Step-by-step explanation:
First, we need to find 10% of 9.50. To do that we must divide 9.50 by 10 or simply move the decimal one place to the left to get 0.95. 0.95 is 10% of 9.50. Now we must ADD 0.95 to 9.50 because Jack is getting a 10% raise. 0.95 + 9.50 = 10.45. Therefore Jack makes $10.45 per hour at his job.
Find the value of b. Round your answer to the nearest hundredth.
Image may not
be drawn to scale.
The value of the tangent segment b is 20.15.
What is the value of side b?The secant-tangent power theorem, also known as the tangent-secant theorem, states that if a tangent and a secant are drawn from a common external point to a circle, then the product of the length of the secant segment and its external part is equal to the square of the length of the tangent segment.
It is expressed as:
( tangent segment )² = External part of the secant segment + Secant segment.
From the diagram:
Tangent segment = WX = b
External part of the secant segment = YX = 14
Secant segment = ZX = 15 + 14 = 29
Plug these values into the above formula and solve for b.
( tangent segment )² = External part of the secant segment + Secant segment.
b² = 14 × 29
b² = 406
b = √406
b = 20.15
Therefore, the value of b is 20.15.
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Help a man out. Its due tomorrow need to get it done.
Answer:
A≈166.28
Step-by-step explanation:
Answer:
A≈166.28
Step-by-step explanation:
A system of linear inequalities is shown
below. Which of the following points are
solutions of the system?
8x + 5y > 40
-6x +2y >_-18
The points (0,10) and (-8,4) are solutions of the system of inequalities.
Define system of inequalityA system of inequalities refers to a set of two or more inequalities that are graphed together on the same coordinate plane. Each inequality in the system represents a region in the plane where the solution to that inequality lies.
To determine if a point is a solution of the system of inequalities, we need to check if it satisfies both inequalities. Let's check each point given:
(4,6):
8(4) + 5(6) = 47 > 40 (true)
-6(4) + 2(6) = -14 ≥ 18 (false)
(9,8):
8(9) + 5(8) = 97 > 40 (true)
-6(9) + 2(8) = -40 ≥ 18 (false)
(0,10):
8(0) + 5(10) = 50 > 40 (true)
-6(0) + 2(10) = 20 ≥ 18 (true)
(5,-6):
8(5) + 5(-6) = 34 > 40 (false)
-6(5) + 2(-6) = -30 ≥ 18 (false)
(-1,-8):
8(-1) + 5(-8) = -44 ≤ 40 (false)
-6(-1) + 2(-8) = 10 ≥ 18 (false)
(-8,4):
8(-8) + 5(4) = -24 ≤ 40 (true)
-6(-8) + 2(4) = 52 ≥ 18 (true)
Therefore, the points (0,10) and (-8,4) are solutions of the system of inequalities.
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Please help -(-4)(-6)-3/5(10+5)/1/3!! I’ll give 30 points.
The solution to the expression -(-4) (-6)-3/5(10+5)/1/3 is 15.
To solve this expression, we need to follow the order of operations, which is also known as PEMDAS (parentheses, exponents, multiplication and division, addition and subtraction):
Start with the parentheses: (-4) (-6)= 24, so we can rewrite the expression as:
24-3/5(15)/(1/3)
Now we need to simplify the expression inside the parentheses by adding 10 and 5, which equals 15. Then we need to simplify the division by multiplying the numerator by the reciprocal of the denominator, which is the same as dividing by 1/3. We can rewrite this expression as:
24-3/5*15*3
Multiply 3/5 and 15 first, then multiply that result by 3: 24-9.
Finally, subtract 9 from 24 to get the final answer: 15.
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Coby's room is a rectangle that measures 10 feet by 8 feet.
use the drop-down menus to complete the statement about the floor of coby's room.
The area of Coby's room is 80 square feet.
The area of a rectangle is given by the product of its length and width. Here, the length of the room is given as 10 feet and the width is given as 8 feet. Therefore, the area of the room is:
Area = Length x Width
Area = 10 feet x 8 feet
Area = 80 square feet
Hence, the area of Coby's room is 80 square feet. It is important to note that when calculating the area of a rectangle, the units of length are multiplied to obtain the unit of area. In this case, the units of length are feet, so the unit of area is square feet.
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can someone please help will give brain
Answer:
y = 9
Step-by-step explanation:
• Parallel lines have equal slopes
calculate the slope m of TU using the slope formula
m = [tex]\frac{y_{2}-y_{1} }{x_{2}-x_{1} }[/tex]
with (x₁, y₁ ) = T(- 7, 4 ) and (x₂, y₂ ) = U(- 2, - 6 )
[tex]m_{TU}[/tex] = [tex]\frac{-6-4}{-2-(-7)}[/tex] = [tex]\frac{-10}{-2+7}[/tex] = [tex]\frac{-10}{5}[/tex] = - 2
now calculate the slope of VW and equate to - 2
with (x₁, y₁ ) = V(8, 7 ) and (x₂, y₂ ) = W(7, y )
[tex]m_{VW}[/tex] = [tex]\frac{y-7}{7-8}[/tex] = [tex]\frac{y-7}{-1}[/tex]
now equating gives
[tex]\frac{y-7}{-1}[/tex] = - 2 ( multiply both sides by - 1 to clear the fraction )
y - 7 = 2 ( add 7 to both sides )
y = 9
What is the probability of spinning a 1 or 2? Write your answer as a fraction AND decimal
Answer:
Probability of spinning 1 or a 2
Fraction=6/8
decimal=0.75
Let me know if it helped
You spin the spinner, flip a coin, then spin the spinner again. Find the probability of not spinning a 5. Flipping tails, then spinning a 5. Write your answer as a traction in simplest form
The probability of not spinning 5 is 81/200.
What is the probability?
The probability of an occurrence is a figure that represents how likely it is that the event will take place. In terms of percentage notation, it is expressed as a number between 0 and 1, or between 0% and 100%. The higher the likelihood, the more likely it is that the event will take place.
Here, we have
Given: You spin the spinner, flip a coin, then spin the spinner again.
We have to find the probability of not spinning a 5. Flipping tails, then spinning a 5.
P(not spinning a 5) = 9/10
P(heads)= 1/2 because there is an equal chance to get heads or tails
P(not spinning a 5)= same as before, 9/10 (independent events).
= [tex]\frac{9}{10}[/tex] × [tex]\frac{1}{2}[/tex] × [tex]\frac{9}{10}[/tex]
= [tex]\frac{81}{200}[/tex]
Hence, the probability of not spinning 5 is 81/200.
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Which expression is equivalent to −0.75(60–32n)–n?
−45+23n
45–23n
−45+31n
−45+15n
The expression that is equivalent to −0.75(60–32n)–n is A. −45+23n.
What is a mathematical expression?A mathematical or algebraic expression is the combination of variables with numbers, constants, and values using algebraic operands, including addition, multiplication, subtraction, and division.
Mathematical expressions do not bear the equal symbol (+) unlike equations.
−0.75(60–32n)–n
Expanding:
-45 + 24n - n
Simplifying:
−45 + 23n
Thus, the equivalent expression is Option A.
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A hot air balloon rising vertically is tracked by an observer located 5 km from the lift-off point. At a certain moment, the angle between the observer's line of sight and the horizontal is π/5 , and it is changing at a rate of 0.4 rad/h. How fast is the balloon rising at this moment? (Round your answer to three decimal places.)
The balloon is rising at a rate of approximately 3.532 km/h at this moment.
We will use the tangent function in trigonometry and the concept of related rates.
Let x be the horizontal distance from the observer to the lift-off point (5 km) and y be the vertical distance from the ground to the balloon. The angle between the observer's line of sight and the horizontal is given as π/5.
We can use the tangent function:
tan(θ) = y/x
At the given moment, x = 5 km and θ = π/5, so:
tan(π/5) = y/5
Now, let's differentiate both sides with respect to time (t):
d(tan(θ))/dt = d(y)/dt / 5
We know that d(θ)/dt = 0.4 rad/h, so we can find d(tan(θ))/dt using the chain rule:
d(tan(θ))/dt = (sec^2(θ)) * d(θ)/dt
d(tan(θ))/dt = (sec^2(π/5)) * 0.4
Now, substitute this back into the first equation and solve for d(y)/dt:
(sec^2(π/5)) * 0.4 = d(y)/dt / 5
d(y)/dt = 5 * (sec^2(π/5)) * 0.4
After calculating the expression, you'll find that d(y)/dt ≈ 3.532 km/h. Thus, the balloon is rising at a rate of approximately 3.532 km/h at this moment.
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What is 43% , 2/5 , 3/7 , and 0. 42 remaining in ascending order ?
Answer:
2/5 < 0.42 < 43% < 3/7
Step-by-step explanation:
Let's convert them all to decimals:
43% = 0.43
2/5 = 0.4
3/7 = 0.428571...
0.42 = 0.42
Now we can arrange them in ascending order:
0.4
0.42
0.43
0.428571...
