Determine whether Rolle’s Theorem can be
applied to on the closed interval If Rolle’s Theorem can
be applied, find all values of in the open interval such
that If Rolle’s Theorem cannot be applied, explain
why not
Answers
Rolle’s Theorem can be applied to the closed interval and the value of x = (12 ±√12)/3
What is Rolle's theorem?Rolle's theorem states that "If a function f is defined in the closed interval [a, b] in such a way that it satisfies the following condition: If is continuous οn [a, b], ii) f is differentiable οn (a, b), and iii) f (a) = f (b), then there exists at leastοne value οf x, let us assume this value to be c, which lies between a and b i.e. (a < c < b) in such a way that f'(c) = 0.".
Here, we have
Given: f(x) = (x-1)(x-2)(x-3), [1,3]
We have to determine whether Rolle’s Theorem can be applied to the closed interval.
This function is continuous in [1, 3] and is differentiable everywhere except at the points x = 1, 2, 3.
This point is in the interval [1, 3], and since Rolle's Theorem requires that the function must be differentiable on the open interval (1, 3).
f(x) = (x-1)(x-2)(x-3)
f'(x) = (x-2)(x-3) + (x-1)(x-3) + (x-1)(x-2)
f'(x) = x² - 5x + 6 + x² - 4x + 3 + x² -3x + 2
f'(x) = 3x² -12x + 11
f'(x) = 0
3x² -12x + 11 = 0
x = (12 ±√12)/3
Hence, Rolle’s Theorem can be applied to the closed interval.
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Related Questions
The graph shown here is the graph of which of the following rational functions?
O A. F(x) = 1/X2
O B. F(x) = 1/(x-2)(x+1)
O C. F(x) = 1(x+2)(x-1)
Answers
The equation of the rational function is F(x) = 1/(x-2)(x+1). Thus, the answer is B. F(x) = 1/(x-2)(x+1).
What is a rational function?It is a type of function that can be expressed in the form f(x) = p(x) / q(x), where p(x) and q(x) are polynomials and q(x) is not equal to zero. Rational functions are used in various areas of mathematics, including calculus and algebra.
The graph of a rational function is the set of points which satisfy the equation of the rational function.
In this case, the equation of the rational function is F(x) = 1/(x-2)(x+1).
The graph of this function is a line which passes through the points (2, 0) and (-1, 0) and has a vertical asymptote at x = 0.
The graph of a rational function can be determined by finding the zeroes, vertical and horizontal asymptotes, and intercepts of the function.
The zeroes of F(x) = 1/(x-2)(x+1) are x = 2 and x = -1. The vertical asymptote is x = 0 and the horizontal asymptote is y = 0.
The intercepts are (2, 0) and (-1, 0).
The graph of F(x) = 1/(x-2)(x+1) is a line which passes through the points (2, 0) and (-1, 0) and has a vertical asymptote at x = 0.
This is different from the graph of F(x) = 1/X² which is a parabola and from the graph of F(x) = 1(x+2)(x-1) which is two lines intersecting at (2, 0) and (-1, 0).
Therefore, the correct answer is B. F(x) = 1/(x-2)(x+1).
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Need help with HW question 5
Answers
The characteristic polynomial of A is λ³ - 4λ² - 3λ + 14. The eigenvalues of A are 2, 1, and 7.
What is rational root theorem?According to the rational root theorem, p is a factor of the constant term and q is a factor of the leading coefficient if a polynomial with integer coefficients has a rational root of p/q, where p and q are integers without any shared factors. As the leading coefficient is 1 and the constant term is 14, the only potential rational roots in this situation are 1, 2, 7, and 14.
The characteristic polynomial of a matrix is given as
det(A - λI)
Here, λ is the eigen value.
det (A - λI)
[tex]\left[\begin{array}{ccc}1 - \lambda&0&1\\2&3 -\lambda&-1\\0&6& \lambda\end{array}\right][/tex]
Taking the determinant we have:
det(A - λI) = (1-λ)[(3-λ)(-λ) - (-1)(6)] - 0[2(-λ) - (-1)(0)] + (-1)[2(6) - (3-λ)(0)]
det(A - λI) = (1-λ)(λ² - 3λ - 6) - (-1)(12) = λ³ - 4λ² - 3λ + 14
Hence, the characteristic polynomial of A is λ³ - 4λ² - 3λ + 14.
The eigen values is the solution of the polynomial.
Using the rational roots of polynomial we have:
λ³ - 4λ² - 3λ + 14. factors as (λ-2)(λ-1)(λ-7)
Hence, the eigenvalues of A are 2, 1, and 7.
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Suppose a pendulum is L meters long. The time, f, in seconds that it takes to swing back and forth once is given by t=2.01 √L. If a pendulum is 13.69
meters long, how long does it take to swing back and forth once?
Round your answer to the nearest tenth.
Answers
Rounded to the nearest tenth, the approximate duration for one swing of a pendulum with a length of 13.69 meters is 7.7 seconds.
We are given that the time, t, in seconds it takes for a pendulum to swing back and forth once is given by:
t = 2.01√L
where L is the length of the pendulum in meters.
We are asked to find the time it takes for a pendulum of length 13.69 meters to swing back and forth once. So we can substitute L = 13.69 into the equation above to get:
t = 2.01√(13.69) ≈ 2.01 × 3.7 ≈ 7.7
Therefore, the time it takes for a pendulum of length 13.69 meters to swing back and forth once is approximately 7.7 seconds (rounded to the nearest tenth).
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What is a rectangular prism?
A right rectangular prism is a box-shaped object, that is, a 3-dimensional solid which has six rectangular faces. Rectangular prisms can also be oblique - leaning to one side - but the side faces are parallelograms, not rectangles. A right rectangular prism is also called a cuboid, box, or rectangular hexahedron. Moreover, "rectangular prism" and "right rectangular prism" are often used interchangeably.
The most common math problems related to this solid are of the type right rectangular prism calc find V or find A, where the letters stand for the Volume and Area, respectively. Let's see the necessary rectangular prism formula and learn how to solve those problems quickly and easily.
How do I find the volume of a rectangular prism?
The rectangular prism volume formula is:
volume = h × w × l,
where h is prism height, w is its width, and l is its length. To calculate the volume of a cardboard box:
Find the box length. For example, it can be equal to 18 in.
Determine its width. Let's say you measured 12 in.
Find out the rectangular prism height. Assume it's 15 in.
Calculate the cuboid volume. Using the rectangular prism volume formula above, we get volume = (18 × 12 × 15) in = 3240 in³.
How do I find the area of a rectangular prism?
The surface area of the cuboid consists of 6 faces - three pairs of parallel rectangles. To find the rectangular prism surface area, add the areas of all faces:
surface_area = 2 × (h × w) + 2 × (h × l) + 2 × (l × w) = 2 × (h × w + h × l + l × w),
where h is prism height, w is its width, and l is its length.
Let's see an example of how to solve the right rectangular prism calc - find A problem. We'll come back to our example with the box and calculate its surface area:
Calculate the rectangular prism surface area. First rectangle area is 15in × 12in = 180in², second 15in × 18in = 270in² and third one 18in × 12in = 216in². Add all three rectangles' areas - it's equal to 666 in² (what a number!) - and finally multiply by 2. The surface area of our cardboard box is 1332in².
Or save yourself some time and use our rectangular prism calculator.
Finally, let's attack the right rectangular prism calc find d (that is, the diagonal) type of problem.
How do I calculate the diagonal of a rectangular prism?
To determine the diagonal of a rectangular prism, apply the formula:
diagonal = √(l² + h² + w²)
where h is prism height, w is its width, and l is its length.
Do you have the feeling that you saw the formula before? Yes, that's possible because this equation resembles the famous one from the Pythagorean theorem.
How to calculate volumes of the other solids?
That rectangular prism was a piece of cake! If you are amazed at how easily you can calculate the volume with our tool, try out the other volume calculators:
triangular prism calculator
cylinder volume calculator
sphere volume calculator
cone volume calculator
pyramid volume calculator
Answers
The diameter of a cylindrical construction pipe is 3ft . If the pipe is 16ft long, what is its volume?
Answers
(I THINK I USED THE RIGHT CALCULATOR)
Answer: V≈113.1
Step-by-step explanation:
V=πr2h=π·1.52·16≈113.09734
Colin invests £980 into his bank account. He receives 6% per year simple interest. How much will Colin have after 7 years? Give your answer to the nearest penny where appropriate.
Answer:
Explanation:
Solving steps:
Answers
Colin have £ 1392 in his bank account after 7 years.
Step-by-step explanation:
Given, Principal = P = £ 980
Rate of interest = R = 6% = = 0.06
Time = T = 7 years
Amount = A = P + S.I
Simple Interest = S. I = PRT = 980 x 0.06 x 7 = £ 411.6
Amount after 7 years = £ 980 + £ 411.6 = £ 1391.6 ≈ £ 1392
Hence, Colin have £ 1392 in his bank account after 7 years.
Valeria is tilling her kitchen wall with two types of traditional hand painted Mexicano tiles. If she uses 56 flower tiles how many sun tiles dose she need to maintain the pattern?
A=28
Answers
Therefore, Aditi's mother requires 342 yellow tiles and 57 red tiles for the floral design on the kitchen wall.
What is proportion?Proportion is a statement that two ratios are equal. A ratio is a comparison of two quantities and is written in the form of a fraction. A proportion can be written in the form of a:b = c:d, where a, b, c, and d are numbers. This means that the ratio of a to b is equal to the ratio of c to d. We can also write a proportion as a fraction, where the numerator is the product of the means (ad) and the denominator is the product of the extremes (bc). Proportions are commonly used in mathematics to solve problems involving ratios and rates.
Here,
Since the floral design is made up of yellow and red hexagonal tiles, we need to determine the proportion of yellow tiles and red tiles in the design. Let's assume that the design consists of x yellow tiles and y red tiles. Each flower in the design consists of 7 hexagonal tiles - 6 yellow tiles and 1 red tile. Therefore, the total number of tiles in the floral design is:
Total number of tiles = 57 flowers x 7 tiles per flower
Total number of tiles = 399 tiles
Since each flower consists of 6 yellow tiles and 1 red tile, the total number of yellow tiles in the design is:
Number of yellow tiles = 6 x Number of flowers
Number of yellow tiles = 6 x 57
Number of yellow tiles = 342
Similarly, the total number of red tiles in the design is:
Number of red tiles = 1 x Number of flowers
Number of red tiles = 1 x 57
Number of red tiles = 57
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Complete question:
Aditi's mother is renovating her kitchen. One of the walls of kitchen is entirely made up of tiles. She wants to have a row of floral designs made up of yellow and red hexagonal tiles in the middle of the wall as shown in the adjoining figure. If there are 57 flowers in a row, find out how many yellow-and red-coloured tiles she requires for that wall.
please help me i need to get these right
Answers
The value of x in the right angled triangle above is 7√3.
What is a right angle triangle?A right-angled triangle is a triangle, that has one of its interior angles equal to 90 degrees or any one angle is a right angle. Therefore, this triangle is also called the right triangle or 90-degree triangle.
To calculate the value of x in the right angle triangle above, we use the formula below.
Formula:
tan∅ = opp./adj.Where:
∅ = Included angle = 30°opp. = Opposite = 7adj. = Adjacent = xtan30° = 1/√3Substitute these values into equation 1 and solve for x
tan30° = 7/xx = 7/tan30°x = 7/(1/√3)x = 7√3
Hence, the value of x is 7√3.
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Q3: The standard form of (7.2 x 10)-(3.4 x 10") a) 2.14 x 10 b) 2.11 x 10 2.12 x 102 d) 2.12 x 10
Answers
The standard form of the expression (7.2 × 10⁵) ÷ (3.4 × 10³) include the following: C. 2.12 x 10².
What is an exponent?In Mathematics, an exponent refers to a mathematical operation that is typically used in conjunction with an algebraic expression in order to raise a quantity to the power of another.
This ultimately implies that, an exponent is represented by the following mathematical expression;
bⁿ
Where:
the variables b and n are numerical values (numbers) or an algebraic expression.n is referred to as a superscript or power.By applying the law of indices for powers of the same base number and division, we have the following:
(7.2 × 10⁵) ÷ (3.4 × 10³
(7.2/3.4) × 10⁵⁻³
2.12 x 10².
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Complete Question:
The standard form of (7.2 × 10⁵) ÷ (3.4 × 10³)
a) 2.14 x 10
b) 2.11 x 10
c. 2.12 x 10²
d) 2.12 x 10
I need to know how to solve this
Answers
Answer:
x = 7
Step-by-step explanation:
4*(2x - 10) = 16
Use Distributive property,
4*2x - 4*10 = 16
8x - 40 = 16
Now, we have to isolate ' x'. To isolate 'x', first Add 40 to both sides,
8x = 16 + 40
8x = 56
Divide both sides by 8,
x = 56 ÷ 8
[tex]\boxed{\bf x = 7}[/tex]
Solve the systems by substitution.
y=4x+6
y = 7x
Answers
Answer:
x = 2
y = 14
Step-by-step explanation:
{y = 4x + 6;
{y = 7x;
.
Let's replace y in the second equation with the value given in the 1st equation:
4x + 6 = 7x
4x - 7x = -6
-3x = -6 / : (-3)
x = 2
x = 2y = 4 × 2 + 6 = 8 + 6 = 14
According to the graph, what is the value of the constant in the equation below?
Answers
According to the graph, the value of the constant in the equation below is 2. The correct option is B.
What is constant in a graph?A constant function in mathematics is one whose (output) value is constant regardless of the input value.
We are given the equation for the height.
On the graphic, height, and width are related. By choosing a point, we can find the value of the constant.
To find a constant in a graph, the width is multiplied by height.
(Height)(Width) = Constant.
Multiply any of the coordinate values shown, and you will see the constant is 2.
3 × 1.5 = 2 × 1 = 8 × 4 = 7 × 3.5 =
we will take only one variable which is 2 x 1 = 2, so the constant is 2
Therefore, the correct option is B, 2.
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I need help on these anybody?
Answers
The complete labels to the parts of the circle such as tangent, diameter, radius and secant are found in the attchment.
What are the parts of a circle?The various parts of a circle are:
Center: The point inside the circle that is equidistant from all points on the circumference.
Circumference: The distance around the circle.
Radius: A line segment from the center of the circle to any point on the circumference.
Diameter: A line segment that passes through the center of the circle and has its endpoints on the circumference.
Chord: A line segment with both endpoints on the circumference.
Tangent: A line that intersects the circle at only one point, called the point of tangency.
Secant: A line that intersects the circle at two points.
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Jessie bought some pizzas and burgers. 3/4 of the food bought were burgers. Jessie bought 18 burgers. How many pizzas and burgers did Jessie buy altogether?
Answers
Answer:
24
Step-by-step explanation:
Total number of food = x
If ¾ of food = number of burgers bought
Then 3/4 times x = 3x/4
3x/4= 18
x = 24
Naomi wants to wrap a gift for her sister. The gift is in a box with a length of 10 inches, a height of 2 inches, and a depth of 3 inches. If Naomi is calculating the minimum amount of wrapping paper to wrap the gift, would she be finding the Volume or Surface Area?
Answers
To calculate the minimum amount of wrapping paper needed to wrap the gift, Naomi would be finding the Surface Area.
The Surface Area of a three-dimensional object is the sum of the areas of all its faces. In this case, the gift is a rectangular prism with dimensions of length 10 inches, height 2 inches, and depth 3 inches.
To wrap the gift, Naomi would need to cover all six faces of the rectangular prism with wrapping paper. The formula for the Surface Area of a rectangular prism is:
Surface Area = 2lw + 2lh + 2wh
where l, h, and w are the length, height, and width of the rectangular prism, respectively.
Substituting the values for the length, height, and depth of the gift, we get:
Surface Area = 2(10 x 2) + 2(10 x 3) + 2(2 x 3)
Surface Area = 20 + 60 + 12
Surface Area = 92 square inches
Therefore, Naomi would need at least 92 square inches of wrapping paper to cover the gift, which is the Surface Area of the rectangular prism.
An inverse variation includes the points (5, 3) and (1, n). Find n.
Write and solve an inverse variation equation to find the answer.
n=
Submit
Answers
Answer:
n = 15
Step-by-step explanation:
the equation for inverse variation is
y = [tex]\frac{k}{x}[/tex] ← k is the constant of variation
to find k use the point (5, 3 ) , that is y = 3 when x = 5 , then
3 = [tex]\frac{k}{5}[/tex] ( multiply both sides by 5 )
15 = k
y = [tex]\frac{15}{x}[/tex] ← equation of variation
substitute (1, n ), that is x = 1, y = n into the equation
n = [tex]\frac{15}{1}[/tex] = 15
At what rate of interest will a sum of 10,000 become 12,544 in 2 years if the interest is compounded annually
Answers
Principal(P) = 10,000
Period (T)= 1 year
Sum amount (A)= 11200
Rate of interest =?
(i) We know that
Interest (1)= 11200-10000=1200
So the rate of interest
R= (1×100)/(PxT)
Substituting the values
R= (1200×100¥/(1000×1)
So we get
R= 12% p.a
Therefore, the rate of interest per annum is 12% p.a
What is the principal square root of -27
Answers
the principle square root of -27 is √5.196.
The equation of the piecewise defined function f(x) is below. What is the value of f(1)?
[x²+1, -4
f(x) = -x², 1
3x, x22
f(1) = -2
f(1) = -1
f(1) = 2
f(1) = 3
Answers
The second piece of the function applies to x = 1, which is f(x) = -x².
Thus, f(1) = -(1)² = -1. The correct answer is f(1) = -1.
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. In simpler terms, a function is a set of rules that takes an input value and produces a corresponding output value.
To find the value of f(1), we need to determine which piece of the function applies to x = 1.
Since 1 is greater than 0 and less than 2, the second piece of the function applies to x = 1, which is f(x) = -x².
Thus, f(1) = -(1)² = -1.
Therefore, the correct answer is f(1) = -1.
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Drag the similar figure into the table. Angle measures are rounded to the nearest integer.
Answers
Triangle (B) is a comparable number that you must drag into the table.
What are angles?
An angle is formed when two straight lines or beams meet at a singular endpoint.
The intersection of two locations is known as the vertex of an arc.
The word "angle" is derived from the Latin word "angulus," which means "corner."
But there are some basic viewpoints.
Combining different shots greatly expands your camera options.
The most confusing of the five angles involves taking a scene from straight overhead.
So, the following:
Using the provided figure, we obtain:
A 60-degree equilateral triangle is depicted in the image.
And as can be seen in figure B, we obtain an equilateral triangle with a 60-degree inclination on each side.
Therefore, triangle (B) is a comparable number that you must drag into the table.
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2477 how to make persantage please if you dont answer you are from ohio
Answers
I am from Ohio
Answer:
247700% I'm not sure
I'm not good I'm maths
pleasee and thank you very much
Answers
The average rate of change between t = 0 and t = 2 is 16 feet/second and the average rate of change between t = 1 and t = 2 is 8 feet/second.
The correct answer choice is option A and C respectively.
What is the average rate of change?The average rate of change between t = 0 and t = 2
At t = 0
h(t) = -16t² + 40t + 6
= -16(0)² + 40(0) + 6
= 0 + 0 + 6
= 6 feet/second
At t = 2
h(t) = -16t² + 40t + 6
= -16(2)² + 40(2) + 6
= -16(4) + 80 + 6
= -64 + 80 + 6
= 22 feet/second
The average rate of change between t = 0 and t = 2 is
22 feet - 6 feet
= 16 feet/second
At t = 1
h(t) = -16t² + 40t + 6
= -16(1)² + 40(1) + 6
= -16 + 40 + 6
= 30 feet/second
Therefore,
The average rate of change between t = 1 and t = 2 is
30 feet - 22 feet
= 8 feet
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I need help on this one
Answers
Answer:
19 degrees
Step-by-step explanation:
hope this helps
Calculate the total volume of Cubic Yards of Concrete needed to fill in the footer. (That is the enclosed frame in the left of the diagram above) (the footer needs to be 2 ft wide and 1 ft deep. (remember there is a 2 ft difference between the outside and the inside of trench)
Answers
The total volume of Cubic Yards of Concrete needed to fill in the footer is approximately 225.19 cubic yards.
To calculate the total volume of Cubic Yards of Concrete needed to fill in the footer, we need to use the formula for the volume of a rectangular prism, which is length x width x height. In this case, the length and height are both 20 feet since that is the size of the enclosed frame in the left of the diagram.
The width of the footer needs to be 2 feet, which means we need to subtract 4 feet (2 feet on each side) from the total width of the frame to get the inside width of the trench, which is 16 feet.
The depth of the footer needs to be 1 foot, which means we need to subtract 1 foot from the height of the frame to get the height of the trench, which is 19 feet.
Using these measurements, we can calculate the volume of the trench by multiplying length (20 feet) x width (16 feet) x height (19 feet) which gives us a total volume of 6,080 cubic feet.
To convert cubic feet to cubic yards, we need to divide the total volume by 27 (since there are 27 cubic feet in a cubic yard). Therefore, the total volume of Cubic Yards of Concrete needed to fill in the footer is approximately 225.19 cubic yards.
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Note the full question is
1) Calculate the total volume of Cubic Yards of Concrete needed to fill in the footer. (That is the enclosed frame in the left of the diagram above) (the footer needs to be 2 ft wide and 1 ft deep. (remember there is a 2 ft difference between the outside and the inside of trench)
2) What is the square footage of the footer trench ?
3) If the footer trench is 2 ft deep, would 10 Cubic-yards of concrete overfill this trench? By how much. ?
An area of a triangle and rectangle
12m
3m
6cm
Area=
Answers
Answer:
The dimensions of the t r including the path are 12+3+3 and 6+3+3 which is 18x12=216 m2
If we subtract the tr area 12x6=72m2
216m2–72m2=144m2
Step-by-step explanation:
principle amount =750 interest rate =5% time = 4 years
how much interest is gained after 4 years ?
Answers
Answer:
Step-by-step explanation:
use formula I = prt
I=Interest P=principal R=rate T=time so Interest is = to 750 x 5% or 0.05 x 4 which is 150 so after 4 years 150 interest is gained
will give brainiest if correct and quick
Answers
Answer: -2， 6
（-2）Because from 3a³ to (1)a³, it needs to subtract 2a³.
(6) Because from -5b³ to b³, it needs to add 6.
Shirley uses pattern blocks to measure the straight angle. Select all the combinations of pattern block angles that Shirley could use to measure the angle
Answers
To measure a straight angle, Shirley could use any combination of pattern blocks that adds up to 180 degrees.
What are the pattern block angle combinations Shirley could use to measure the angle?
One hexagon (120 degrees) and three triangles (60 degrees each): 120 + 60 + 60 + 60 = 300 degrees (not a straight angle)Two triangles (60 degrees each) and two hexagons (120 degrees each): 60 + 60 + 120 + 120 = 360 degrees (a full circle, which includes a straight angle)One trapezoid (270 degrees) and one triangle (60 degrees): 270 + 60 = 330 degrees (not a straight angle)One rhombus (120 degrees) and two trapezoids (270 degrees each): 120 + 270 + 270 = 660 degrees (not a straight angle)Three rhombuses (120 degrees each) and one triangle (60 degrees): 120 + 120 + 120 + 60 = 420 degrees (not a straight angle)Six triangles (60 degrees each): 60 + 60 + 60 + 60 + 60 + 60 = 360 degrees (a full circle, which includes a straight angle)As a result, the pattern block angle combinations that Shirley could use to measure the angle are:
Two triangles (60 degrees each) and two hexagons (120 degrees each)Six triangles (60 degrees each)
Incomplete question provided, below is the complete question -
Shirley measures the straight angle using pattern blocks. Choose every possible combination of angle possibilities for pattern blocks Shirley could use to calculate the angle.
# The tan pattern block has six minor angles.
# On the red pattern block, there are two angles: one large and one small.
# On the red pattern block, there is one large angle and three small angles.
# The tan pattern block has four tiny angles, and the red pattern block has one tiny angle.
# The red pattern block has two sharp angles.
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Three hundred consumers were surveyed about a new brand of snack food, Crunchicles. Their age groups and preferences are given in the table.
Answers
a) the probability that the consumer is 18-24 years of age, given that he/she dislikes Crunchicles, is 0.0889 or approximately 0.089.
b) the probability that the selected consumer dislikes Crunchicles is 0.35 or 35%.
c) the probability that the selected consumer dislikes Crunchicles is 0.35 or 35%.
d) the probability that the selected consumer is 35-55 years old or likes Crunchicles is 0.44 or 44%.
What is probability?
Probability is a measure of the likelihood or chance of an event occurring. It is a number between 0 and 1, with 0 representing an impossible event and 1 representing a certain event. The probability of an event is calculated by dividing the number of ways the event can occur by the total number of possible outcomes.
To solve these problems, we need to use the information given in the table and apply the rules of conditional probability and total probability.
What is the probability that the consumer is 18-24 years of age, given that he/she dislikes Crunchicles?
We want to find P(18-24 | disliked). By Bayes' theorem, we have:
P(18-24 | disliked) = P(disliked | 18-24) * P(18-24) / P(disliked)
We know that P(disliked | 18-24) = 17/132, P(18-24) = 92/300, and P(disliked) = 150/300. Therefore,
P(18-24 | disliked) = (17/132) * (92/300) / (150/300) = 0.0889
So the probability that the consumer is 18-24 years of age, given that he/she dislikes Crunchicles, is 0.0889 or approximately 0.089.
What is the probability that the selected consumer dislikes Crunchicles?
We want to find P(disliked). We know that the total number of consumers surveyed is 300 and the number of consumers who disliked Crunchicles is 105. Therefore,
P(disliked) = 105/300 = 0.35
So the probability that the selected consumer dislikes Crunchicles is 0.35 or 35%.
What is the probability that the selected consumer is 35-55 years old or likes Crunchicles?
We want to find P(35-55 or likes). By the rule of total probability, we have:
P(35-55 or likes) = P(35-55) + P(likes)
We know that P(35-55) = 32/300 and P(likes) = 92/300. Therefore,
P(35-55 or likes) = 32/300 + 92/300 = 0.44
So the probability that the selected consumer is 35-55 years old or likes Crunchicles is 0.44 or 44%.
Hence, a) the probability that the consumer is 18-24 years of age, given that he/she dislikes Crunchicles, is 0.0889 or approximately 0.089.
b) the probability that the selected consumer dislikes Crunchicles is 0.35 or 35%.
c) the probability that the selected consumer dislikes Crunchicles is 0.35 or 35%.
d) the probability that the selected consumer is 35-55 years old or likes Crunchicles is 0.44 or 44%.
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Answers
The total surface area of the square pyramid is 39 ft².
What is the total surface area?The total surface area of a square pyramid can be determined by adding the area of the lateral surfaces and the base together. A square pyramid has four lateral sides and one base.
Total surface area= area of the base + area of the lateral sides
Area of the base = area of a square = length² = 3² = 9 ft²
Area of the lateral sides = 2 x base x height
2 x 3 x 5 = 30 ft²
Total surface area = 30 + 9 = 39 ft²
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