Answers
For the given quadratic function the features are:
Domain: (-∞, ∞): Range :[-1, ∞)Vertex: (-3, -1)Explain about the quadratic function?A parabola, a U-shaped curve, is the shape of a quadratic function's graph.
The graph's vertex, which is an extreme point, is one of its key characteristics. The vertex, or lowest point on the graph or minimal value of a quadratic function, is where the parabola will open up.There are three characteristics that all quadratic functions share:
A quadratic function's graph is always a parabola with an end behaviour that is either upward or downward; its domain will be all real numbers; and its vertex is really the lowest point once the parabola opens upwards.Thus, for the given quadratic function the features are:
Domain: (-∞, ∞): Range :[-1, ∞)Vertex: (-3, -1)Know more about quadratic function
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Related Questions
The Moondance Riding Academy held its annual horse show for 3 days. The total amount collected in entry fees for the 3 days was $1,450. The amount collected, in dollars, is shown for each of the 3 days in the bar graph below: Approximately what percent of the money collected from entry fees over the 3 days was collected on Day 2?
pleas help me
Answers
Answer:
To find the percentage of the money collected from entry fees over the 3 days that was collected on Day 2, we need to first find the total amount collected for all three days.
Total amount collected = $750 + $400 + $300 = $1450
Then, we need to find what fraction of the total was collected on Day 2:
Amount collected on Day 2 = $400
Fraction of total collected on Day 2 = $400 / $1450
Finally, we convert the fraction to a percentage:
Fraction of total collected on Day 2 = $400 / $1450 = 0.2759
Percentage of total collected on Day 2 = 0.2759 x 100% = 27.59%
Therefore, approximately 27.59% of the money collected from entry fees over the 3 days was collected on Day 2.
For a function f(x)f(x) and a particular input value x=ax=a, then we may write the difference quotient as
f(a+h)−f(a)/h
where h≠0
Now, let f(x)=x^3−15x and consider the input value a=3. We could now write the difference quotient as
f(3+h)−f(3)/h
where h≠0.
Use this difference quotient to calculate the average rate of change of f(x) from x=3 to x=3+h for the following particular values of h.
When h=0.2 the average rate of change of f(x) is __________ .
When h=0.1, the average rate of change of f(x) is ____________ .
When h=0.01, the average rate of change of f(x) is ____________ .
When h=−0.01, the average rate of change of f(x) is _____________ .
When h=−0.1, the average rate of change of f(x) is ______________ .
When h=−0.2, the average rate of change of f(x) is ______________ .
Answers
The average rate of change of f(x) for the given values of h are:
When h=0.2, the average rate of change of f(x) is 112.84.
When h=0.1, the average rate of change of f(x) is 206.41.
When h=0.01, the average rate of change of f(x) is 1850.5601.
When h=−0.01, the average rate of change of f(x) is -1807.5399.
When h=−0.1, the average rate of change of f(x) is -171.39.
When h=−0.2, the average rate of change of f(x) is -80.76.
To calculate the average rate of change of f(x) for the given values of h, we need to substitute the values of h into the difference quotient and simplify.
When h=0.2:
f(3+0.2)−f(3)/0.2 = [(3.2^3−15(3.2)) - (3^3−15(3))]/0.2 = (32.768 - 10.2) - (27 - 45)/0.2 = 22.568/0.2 = 112.84
When h=0.1:
f(3+0.1)−f(3)/0.1 = [(3.1^3−15(3.1)) - (3^3−15(3))]/0.1 = (29.791 - 9.15) - (27 - 45)/0.1 = 20.641/0.1 = 206.41
When h=0.01:
f(3+0.01)−f(3)/0.01 = [(3.01^3−15(3.01)) - (3^3−15(3))]/0.01 = (27.270601 - 8.765) - (27 - 45)/0.01 = 18.505601/0.01 = 1850.5601
When h=−0.01:
f(3+(-0.01))−f(3)/(-0.01) = [(2.99^3−15(2.99)) - (3^3−15(3))]/(-0.01) = (26.730399 - 8.655) - (27 - 45)/(-0.01) = 18.075399/(-0.01) = -1807.5399
When h=−0.1:
f(3+(-0.1))−f(3)/(-0.1) = [(2.9^3−15(2.9)) - (3^3−15(3))]/(-0.1) = (24.389 - 8.25) - (27 - 45)/(-0.1) = 17.139/(-0.1) = -171.39
When h=−0.2:
f(3+(-0.2))−f(3)/(-0.2) = [(2.8^3−15(2.8)) - (3^3−15(3))]/(-0.2) = (21.952 - 7.8) - (27 - 45)/(-0.2) = 16.152/(-0.2) = -80.76
Therefore, the average rate of change of f(x) for the given values of h are:
When h=0.2, value is 112.84.
When h=0.1, value is 206.41.
When h=0.01, value is 1850.5601.
When h=−0.01, value is -1807.5399.
When h=−0.1, value is -171.39.
When h=−0.2, value is -80.76.
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4. Consider an isosceles triangle A ABC with base BC and ABC = 75°. We constructs points M on AC and N, P on AB so that BC = MB = PM and MN || BC. (a) Find BMN. (b) Find AMP. Justify your claims. A P
Answers
Answer:4. (a) BMN = 75°(b) AMP = 105°Explanation:Given that ABC is an isosceles triangle with base BC, and ABC = 75°. Also, BC = MB = PM and MN || BC.(a) To find BMN, we can use the fact that MN || BC. This means that ∠BMN = ∠ABC = 75°. Therefore, BMN = 75°.(b) To find AMP, we can use the fact that BC = MB = PM. This means that triangle BMP is an isosceles triangle with base MP. Since BMN = 75°, we can find ∠BMP by using the fact that the sum of the angles in a triangle is 180°:∠BMP = (180° - 75°)/2 = 52.5°Now, we can find AMP by using the fact that the sum of the angles in a triangle is 180°:AMP = 180° - 75° - 52.5° = 52.5°Therefore, AMP = 105°.
(a) The angle BMN is 30°.
(b) The angle AMP is 15°.
The total angle of a triangle is 180°.
How to find the angle in a triangle?We have an isosceles triangle ABC with base BC and ∠ABC is 75°. We construct point M on AC and N, P on AB. It makes BC = MB = PM and MN || BC.
Since ABC is an isosceles triangle, ∠ACB is also 75°. See the picture attached!
Since BC = MB, the triangle MBC is isosceles with base MC. So, BMC is 75°.
The angles of triangle MBC will sum up to 180°. So,
∠MBC = 180 - (75 + 75) = 30°
The angle BMN and MBC is alternate interior angles, so
∠BMN = ∠MBC = 30°
Now, we have ∠NBM = 75 - 30 = 45°. Since MB = PM, it makes BMP an isosceles triangle. The angle BPM will be the same with PBM. It is 45°.
The angle PMN will be
= 180 - (45 + 45 + 30)
= 60°
The angle CMB, BMN, NMP, and AMP are supplementary. So, the angle AMP is
= 180 - (75 + 30 + 60)
= 15°
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write the word form of decimal 0.609.
Answers
Answer: six hundred nine thousandths
Step-by-step explanation:
For the following exercises, use the properties of logarithms to expand each logarithm as much as possible. Rewrite each expression as a sum, difference, or product of logs. 15.log(z19x1319)16.ln(b−40a−2)17.log(x3y−4)18.ln(y1−yy)
Answers
The properties of logarithms, we can expand each expression and rewrite it as a sum, difference, or product of logs. This makes it easier to simplify and solve the expression.
The properties of logarithms can be used to expand each logarithm as much as possible. These properties include the product property, the quotient property, and the power property. Using these properties, we can rewrite each expression as a sum, difference, or product of logs.
15. log(z^19*x^13*19)
= log(z^19) + log(x^13) + log(19)
= 19*log(z) + 13*log(x) + log(19)
16. ln(b^−4*0*a^−2)
= ln(b^-4) + ln(0) + ln(a^-2)
= -4*ln(b) + ln(0) + -2*ln(a)
17. log(x^3*y^-4)
= log(x^3) + log(y^-4)
= 3*log(x) + -4*log(y)
18. ln(y^1*−y*y)
= ln(y^1) + ln(-y) + ln(y)
= 1*ln(y) + ln(-y) + ln(y)
By using the properties of logarithms, we can expand each expression and rewrite it as a sum, difference, or product of logs. This makes it easier to simplify and solve the expression.
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Mariana buys bottles of orange juice at the corner store. Assume each bottle of juice is the same price. The proportional relationship between the number of juice bottles bought, j, and the total cost in dollars and cents, c, can be represented by the equation
�
=
0. 8
�
c=0. 8j. What is the cost in dollars and cents of each bottle of juice?
Answers
By evaluating the function in j = 1, we conclude that each bottle costs 80 cents.
We know that the total cost in dollars can be represented by the linear equation:
C(j) = 0.8*j
Where C is the cost, and j is the number of juice bottles bought.
Then to get the cost of each bottle, we can evaluate the function in 1, which means buying only one.
C(1) = 0.8*1 = 0.8
So each bottle costs $0.80 (or 80 cents).
In mathematics, a function is a relationship between a set of inputs and a set of possible outputs, such that each input is associated with a unique output. Functions are represented by mathematical expressions or equations that specify how the input values are transformed into output values. For example, the function f(x) = x^2 takes an input value x and produces an output value equal to x squared.
Functions play a crucial role in many areas of mathematics, science, and engineering. They can be used to model complex systems, analyze data, and solve problems. Functions are also used in computer programming, where they are used to encapsulate a specific set of operations that can be reused throughout a program.
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Write a polynomial of least degree with integral coefficients that has the given zeros.
1.0, 5, 2
2.-3,-5,3,-1
5. 2 mult.2, -4
3.-5, 2 + i
4.-1, 2i
Answers
The polynomial expressions are P(x) = x(x - 5)(x - 2), P(x) = (x + 3)(x + 5)(x - 3)(x + 1), P(x) = (x + 5)(x² - 4x + 5) and P(x) = (x + 5)(x² + 4)
How to determine the polynomial expressionsZeros (1): 0, 5, 2
This means that
Roots: x = 0, x = 5 and x = 2
The equation can be represented as
P(x) = (x - root)
So, we have
P(x) = (x - 0)(x - 5)(x - 2)
Evaluate
P(x) = x(x - 5)(x - 2)
Zeros (2): 0, 5, 2
Here, we have
Roots: x = -3, x = -5, x = 3 and x = -1
Using the form in (a), we have
P(x) = (x + 3)(x + 5)(x - 3)(x + 1)
Zeros (3): 0, 5, 2
Here, we have
Roots: x = -5, x = 2 + i
Using the form used abov]e, we have
P(x) = (x + 5)(x - 2 - i)(x - 2 + i)
Expand
P(x) = (x + 5)((x - 2)² + 1)
So, we have
P(x) = (x + 5)(x² - 4x + 5)
Zeros (4): -1 and 2i
Here, we have
Roots: x = -1, x = 2i
Using the form used abov]e, we have
P(x) = (x + 1)(x - 2i)(x + 2i)
Expand
P(x) = (x + 5)(x² + 4)
Zeros (5):2 mult.2, -4
The zeros here are not clear
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This diagram is a genetic map that shows the relative positions of four alleles
along a chromosome.
Chromosome
H
G
k
p
Which two alleles have the highest probability of being inherited together?
A. G and K
B. H and K
C. G and P
D. H and P
Answers
The two alleles that have the highest probability of being inherited together are A. G and k.
Why would certain alleles have a greater probability of being inherited together?Alleles that are physically close to each other on a chromosome have a higher probability of being inherited together, a phenomenon known as linkage. The closer two alleles are to each other on a chromosome, the less likely it is that they will be separated by a recombination event during meiosis.
The degree of linkage between two alleles is determined by their relative physical distance on the chromosome. The closer two alleles are, the higher the probability that they will be inherited together.
G and k are the closest alleles to each other on the genetic map and so have a higher probability of being inherited together.
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Please help! Will mark brainliest!
Answers
Different triangles with the same areas include:
Right angle triangle: base = 6, height = 4
Acute triangle: base = 8, height = 3
Obtuse triangle: base = 16, height = 1.5
How to create triangles with the same areas?To create a right angle triangle, an acute triangle, and an obtuse triangle that all have the same area, we can use the following dimensions:
Right triangle: base = 6, height = 4
Acute triangle: base = 8, height = 3
Obtuse triangle: base = 16, height = 1.5
We can calculate the area of each triangle using the formula A = 1/2 (base x height):
Right triangle: A = 1/2 (6 x 4) = 12
Acute triangle: A = 1/2 (8 x 3) = 12
Obtuse triangle: A = 1/2 (16 x 1.5) = 12
Therefore, we can see that all three triangles have the same area, which is 12 square units. We know this because they all have the same base-to-height ratios, which means they have the same "steepness" or "slope". This means that the area of each triangle is proportional to its base and height, and since they all have the same ratio of base to height, they will all have the same area.
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Theorem proof?
Thanks
Answers
Triangle AXC is congruent to triangle AYB by the SAS Congruence Criteria.
Describe Congruency of triangle?In geometry, two triangles are said to be congruent if they have the same shape and size. More specifically, two triangles are congruent if all three sides and all three angles of one triangle are equal to the corresponding sides and angles of the other triangle. When two triangles are congruent, they can be superimposed on top of each other, and all corresponding parts of the triangles will match exactly.
There are several ways to prove that two triangles are congruent, including using the side-angle-side (SAS) theorem, the angle-side-angle (ASA) theorem, the side-side-side (SSS) theorem, and the hypotenuse-leg (HL) theorem.
Given that |AB|=|CD| and triangle ABC is isosceles.
We need to prove that triangle AXC = triangle AYB.
To do so, we will use the Side-Angle-Side (SAS) Congruence Criteria.
AC = BC (Given)
∠ACX = ∠BCY (Vertical Angles)
∠CAX = ∠CBY (Base Angles of Isosceles Triangle)
Therefore, triangle AXC is congruent to triangle AYB by the SAS Congruence Criteria.
Hence, proved.
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Decide whether exch of the following sets is (a) cokntable or (b) unorumtable : (Note : a finite set is considered to be coumtable).
{x∈R:2
{x∈Q:∣x∣≥2}
{x∈Q:∣x∣<4}
Answers
The first set is countable, the second set is also countable. The third set is also countable.
(a) The set {x∈R:2 is countable because it is finite.
(b) The set {x∈Q:∣x∣≥2} is countable because it is a subset of the rational numbers, which are countable.
(c) The set {x∈Q:∣x∣<4} is countable because it is a subset of the rational numbers, which are countable.
In Maths, sets are referred as a collection of well-defined objects or elements. A set is usually represented by a capital letter symbol and the total number of elements in the finite set is basically represented as the cardinal number of a set in a curly bracket.
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$18,000 is deposited for 9 years in an account earning 4% interest. (Round your answers to two decimal places.)
(a) Calculate the future value of the investment if interest is compounded semiannually.
$ _____
(b) Calculate the future value if interest is compounded quarterly.
$ _____
(c) How much greater is the future value of the investment when the interest is compounded quarterly?
$ _____
Answers
(a) The future value of the investment if interest is compounded semiannually is $24,840.28.
(b) The future value of the investment if interest is compounded quarterly is $25,025.68.
(c) The future value of the investment is $185.40 greater when the interest is compounded quarterly.
The future value of an investment can be calculated using the formula FV = P(1 + r/n)^(nt), where FV is the future value, P is the principal amount deposited, r is the annual interest rate, n is the number of times interest is compounded per year, and t is the number of years the investment is held.
If the interest is compounded semiannually, then n = 2. Plugging in the given values, we get:
FV = 18000(1 + 0.04/2)^(2*9)
FV = 18000(1.02)^18
FV = $24,840.28
So the future value of the investment if interest is compounded semiannually is $24,840.28.
If the interest is compounded quarterly, then n = 4. Plugging in the given values, we get:
FV = 18000(1 + 0.04/4)^(4*9)
FV = 18000(1.01)^36
FV = $25,025.68
So the future value of the investment if interest is compounded quarterly is $25,025.68.
To find out how much greater the future value of the investment is when the interest is compounded quarterly, we can subtract the future value when interest is compounded semiannually from the future value when interest is compounded quarterly:
$25,025.68 - $24,840.28 = $185.40
So the future value of the investment is $185.40 greater when the interest is compounded quarterly.
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Can someone help me write a proof for this problem
Answers
The solution is, the triangle ABC and DBC are congruent, by the Side-Angle-Side Triangle Congruence Theorem.
What is triangle?A triangle is a polygon with three edges and three vertices. It is one of the basic shapes in geometry. A triangle with vertices A, B, and C is denoted \triangle ABC. In Euclidean geometry, any three points, when non-collinear, determine a unique triangle and simultaneously, a unique plane.
here, we have,
As we can seen from the given figure that two triangles are given.
The triangles ABC and BDC are :
AB = CD
AC = BD
BC = BC
Hence using the SSS theorem of the triangle, the triangle ABC and DBC are congruent, Thus the angle A is equal to angle D.
Hence the option D and E are correct. i.e. congruent by the Side-Angle-Side Triangle Congruence Theorem.
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3/a times what equals 1
Answers
Answer: 3
Step-by-step explanation: try to solve for a by itself by multiplying it on both sides that give you 3 = 1*a. that simply just means that a is equal to 3, so you substitute three to the a variable that will give 3/3 that's equal to 1.
find the volume of the rectangular pyramid
Answers
Answer:
Refer to pic...........
what is x^-2y(x^3y^5)^3 in simplest form
Answers
Find the measurements of X
pt. 2
Answers
The measure of the angle x is 50 degrees
How to determine the angles
In order to determine the value measure of the angle at the center, we need to know that;
The angle subtended by an arc at the center is said to be twice the angle subtended at the circumference.
In other words, we can say that the angle at the center is double the angle at the circumference.
This is represented as;
x = 2x
Such that '2x' is the angle at the circumference
From the image shown, we have that;
x = 2(25)
multiply the value
x = 50 degrees
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The measures of the angles of a triangle are shown in the figure below. Solve for x.
Answers
The value of x will be 6° according to the remaining angles values in the diagram.
What is a Triangle?
With three sides, three angles, and three vertices, a triangle is a closed, two-dimensional object. A polygon also includes a triangle.
How are the sum of triangles calculated in a triangle?
In a triangle, the total interior angles are supplementary. In other words, the sum of a triangle's inner angle measurements is 180°. Hence, we can write the triangle sum theorem's formula as A + B + C = 180° for the triangle ABC.
Now according to the question
=> 40+110+8x-18=180 (Theorem)
=> 150+8x-18=180
=> 8x=180-150+18
=> 8x=48
=> x=6°
The value of x will be : 6°
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Layla goes out to lunch. The bill, before tax and tip, was $16. 90. A sales tax of 6% was added on. Layla tipped 14% on the amount after the sales tax was added. How much tip did she leave? Round to the nearest cent
Answers
Answer:
2.51
Step-by-step explanation:
Layla goes out to lunch. The bill, before tax and tip, was $16.90. A sales tax of 6% was added on. Layla tipped 14% on the amount after the sales tax was added. How much tip did she leave? Round to the nearest cent.
Find the amount after sales tax was added:
Find the amount after sales tax was added:
bill
+
sales tax
=
bill+sales tax=
100
%
+
6
%
100%+6%
=
=
106
%
106%
amount after tax
=
amount after tax=
106
%
of
bill
106% of bill
=
=
1.06
×
$
16.90
1.06×$16.90
106% = 1.06
=
=
$
17.914
$17.914
Click to see alternative method
Find the tip:
Find the tip:
tip
=
tip=
14
%
of
amount after tax
14% of amount after tax
=
=
0.14
×
$
17.914
0.14×$17.914
14% = 0.14
=
=
$
2.50796
$2.50796
≈
≈
$
2.51
$2.51
Round to the nearest cent
Layla left a $2.51 tip.
Layla left a $2.51 tip.
If 5 dogs share equally a bag of dog treats, each dog gets 24 treats.
Suppose 8 dogs share equally the bag of treats. How many treats
does each dog get? (EXPLAIN)
Answers
5 • 24 = 120
If 5 dogs equally share the bag of dog treats, there is a total of 120 treats. We now can apply this to the scenario of 8 dogs distributing:
120 / 8 = 15
If 8 dogs share equally the bag of treats, each dog will receive 15 treats.
Find the distance between the two points.
✓ [?]
(-1,-2)
(3,-1)
Enter the number that
goes beneath the
radical symbol.
Answers
Answer: the number that goes beneath the radical symbol is 17
Olaf lives in a dorm in a tiny room that he shares with three others. He wants to live off campus next year with his friends, but he needs more money from his parents to finance the move. He decides to build a scale model of his dorm room so that when he goes home for break, he can show his parents the cramped conditions he lives in. He decides to let 1 inch represent 30 inches of the actual lengths in the room. His desk is a right rectangular prism 40 inches high, 36 inches long, and 20 inches wide. He decides that his scaled desk should be 1 1/3 inches high, but a roommate says it should be 10 inches high. Who is right and why? What are the other dimensions of the scaled desk?
Answers
The roommate is wrong. The scaled desk should be 1 1/3 inches high, as per Olaf's initial scaling of 1 inch representing 30 inches of actual length in the room. The other dimensions of the scaled desk would be 12 inches long and 6 2/3 inches wide.
When Olaf decided to let 1 inch represent 30 inches of actual length in the room, he created a scale factor of 1:30. This means that for every inch in the scale model, there are 30 inches in the actual room. Using this scale factor, the scaled desk height should be 40/30 = 4/3 inches. Therefore, the roommate's suggestion of 10 inches high is incorrect.
To find the other dimensions of the scaled desk, we need to apply the same scale factor of 1:30. The scaled length would be 36/30 = 12 inches, and the scaled width would be 20/30 = 2/3 inches. However, it's easier to work with whole numbers, so we can multiply both dimensions by 10 to get 120 inches long and 6 2/3 inches wide, which is equivalent to 12 inches long and 2/3 inches wide in the scaled model.
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A paper company needs to ship paper to large printing business. The paper will be shipped in small boxes and large boxes. The volume of each small box is 7 cubic feet and the volume of each large box is 14 cubit feet. A total of 21 boxes of paper were shipped with a combined volume of 252 cubic feet. write a system of equations that could be determined the number of small boxes shipped in the number of large boxes shipped. Defined the variables that you use to write the system.
Answers
Answer: Let's use the variables x and y to represent the number of small boxes and large boxes shipped, respectively.
The system of equations that could be used to determine the number of each type of box is:
x + y = 21 (equation 1)
7x + 14y = 252 (equation 2)
In equation 1, the left side represents the total number of boxes shipped, which is the sum of the number of small boxes (x) and the number of large boxes (y). The right side of the equation represents the total number of boxes shipped, which is given as 21.
In equation 2, the left side represents the total volume of paper shipped, which is the sum of the volume of all the small boxes (7 cubic feet per box) and the volume of all the large boxes (14 cubic feet per box), multiplied by the number of boxes shipped. The right side of the equation represents the total volume of paper shipped, which is given as 252 cubic feet.
These two equations can be used to solve for x and y, which represent the number of small boxes and large boxes shipped, respectively.
Step-by-step explanation:
Ashley reads 64 pages of her book on Monday, 56 pages on Tuesday. She reads a total of 180 pages on Monday through Wednesday. How many pages did she read on Wednesday?
Answers
Answer:
60 pages
Step-by-step explanation:
64 + 56 = 120
180-120 = 60 pages
where does fission regularly occur and what kind of energy is produced
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Mrs. Andretti is having new drapes made for her living room. The cost of the fabric is $15 per yard. The fee to make and hang the drapes is $250. She uses the expression 15x + 250 to calculate the total cost of the drapes. Mrs. Andretti states that x represents the total cost of the fabric. Is she correct? Responses Yes, x represents the total cost of the fabric. Yes, x represents the total cost of the fabric. No, x represents the total cost of the drapes. No, x represents the total cost of the drapes. No, x represents the total yards of fabric used. No, x represents the total yards of fabric used. No, x represents the total amount of fabric she already has.
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Mrs. Andretti is right, of course. In the equation 15x + 250, x stands for the fabric's overall expense.
what is expression ?An expression in mathematics is a collection of numbers, variables, and operations that can be evaluated to create a value, including addition, subtraction, multiplication, and division. Expressions can be straightforward—like 3 + 4 or 5x—or complex—involving numerous factors and operations. From basic arithmetic to intricate functions and equations, a broad range of mathematical concepts can be represented by expressions.
given
The fabric's overall cost is indicated by the value x.
No, x stands for the drapes' total expense.
No, x is the overall number of yards of fabric used.
Mrs. Andretti is right, of course. In the equation 15x + 250, x stands for the fabric's overall expense.
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Gavin wants to buy a skateboard that sells for $49.99. An advertisement says that next week the skateboard will be on sale for $42.50 how much will Gavin save if he waits until next week to buy the skateboard.
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Answer:
$7.49
Step-by-step explanation:
49.99-42.50=7.49
3. Let \( A=\left[\begin{array}{cc}a & b \\ a & b\end{array}\right] \), where \( a \neq 0 \). Find rank \( A \). Also, find a basis of the null space and a basis of the row space of \( A \); and show
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rank(A) = 1
Given that A = $\left[\begin{array}{cc}a & b \\ a & b\end{array}\right]$, where $a \neq 0$.Let's find the rank of A:Rank(A) = number of leading 1's in the row echelon form of A.= $\left[\begin{array}{cc}a & b \\ a & b\end{array}\right]$=> $\left[\begin{array}{cc}a & b \\ 0 & 0\end{array}\right]$There are two leading 1's in row 1. Hence, rank(A) = 1.Now, let's find the basis of the null space of A:Null space of A is the set of all solutions to the equation $Ax=0$.So, $Ax$ = $\left[\begin{array}{cc}a & b \\ a & b\end{array}\right]$ $\left[\begin{array}{c}x \\ y\end{array}\right]$= $\left[\begin{array}{c}ax+by \\ ax+by\end{array}\right]$= $\left[\begin{array}{c}(a+b)x \\ (a+b)y\end{array}\right]$= $\left[\begin{array}{c}0 \\ 0\end{array}\right]$=> (a + b) x = 0 and (a + b) y = 0=> $x = \frac{-b}{a}y$=> $\left[\begin{array}{c}x \\ y\end{array}\right]$= $\left[\begin{array}{c}\frac{-b}{a}y \\ y\end{array}\right]$= $y \left[\begin{array}{c}\frac{-b}{a} \\ 1\end{array}\right]$=> Null space of A = $\{y \left[\begin{array}{c}\frac{-b}{a} \\ 1\end{array}\right] | y \in R \}$Let's take a = 1, b = 2. So, the null space of A = $\{y \left[\begin{array}{c}-2 \\ 1\end{array}\right] | y \in R \}$Therefore, the basis of the null space of A = $\left[\begin{array}{c}-2 \\ 1\end{array}\right]$Now, let's find the basis of the row space of A:Row space of A is the span of the rows of A.=> Row space of A = span $\left\{\left[\begin{array}{cc}a & b\end{array}\right]\right\}$Since rank(A) = 1, there is only 1 non-zero row. Therefore, a basis of the row space of A is $\left\{\left[\begin{array}{cc}a & b\end{array}\right]\right\}$.
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a Theorem : The edge of regression of the polar developable of a space cuive is the locus of the centre of spherical curvatures. D.
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The theorem you mentioned is known as Meusnier's theorem. The edge of regression of the polar developable of a space curve is the locus of the center of spherical curvatures.
It states that the locus of the center of spherical curvature of a space curve is the edge of regression of the polar developable of the curve. In other words, the center of curvature of a curve at any point lies on the edge of regression of the polar developable curve.
This theorem is used to describe the curvature of a curve in three-dimensional space. It is named after the French mathematician Jean Baptiste Meusnier, who first discovered it in 1776.
In mathematical terms, the theorem can be expressed as follows:
Let C be a space curve with curvature k and torsion T. Let P be a point on C and N be the normal vector at P. Then, the center of curvature of C at P is given by:
O = P + (1/k)N
The edge of regression of the polar developable of C is the locus of the point O as P varies along C.
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The number of weekly flu tests increases by a rate of about 32% each week. During the initial week of flu season, 33flu tests were administered. The situation can be modeled by f(x)=33(0.32)^(x) CORRECT AND REWRITE THE FALSE STATEMENT:
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The correct statement is f(x)=33(1.32)^(x). .
About mathematical statementThe original statement is false because it does not accurately represent the increase in the number of weekly flu tests. The original statement suggests that the number of tests decreases by 32% each week, which is not the case.
To accurately represent the increase in the number of weekly flu tests, we need to use the growth factor of 1.32, which is equivalent to 100% + 32%. This means that the number of weekly flu tests increases by 32% each week, as stated in the problem.
Therefore, the correct statement is f(x)=33(1.32)^(x).
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