Answer: The value of the limit of the function may not be equal to the value of the function evaluated at a particular x-value. In other words, just because a function takes on a certain value at a particular x-value doesn't mean that the limit of the function at that x-value is equal to that value. To determine the limit of the function, you may need to use different methods such as L'Hopital's Rule or other limit laws. Additionally, it's important to keep in mind that the limit of the function may not exist, in which case it wouldn't be equal to any specific value.
Step-by-step explanation:
If the information in this pictogram was displayed in a pie chart what would the central angle of the Dexter representing moderate conditions be
The central angle of the pie chart representing the moderate conditions is 80°
What is central angle?Central angle is the angle subtended by an arc of a circle at the center of a circle. The radius vectors form the arms of the central angle.
Given that, a pictogram which was displayed in a pie chart, we need to find the central angle of the pie chart,
Sea conditions is shown,
Calm = 3 blue dots
Moderate = 2 and a half dot
Rough = 5 and 3 parts of the dot
Each dot = 4 points
Therefore,
Calm = 3×4 = 12
Moderate = 2×4 + 1/2 × 4 = 10
Rough = 5×4 + 3/4 ×4 = 23
The central angle = 10 / 12+10+23 × 360° = 80°
Hence, the central angle of the pie chart representing the moderate conditions is 80°
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The complete question is:-
If the information in this pictogram was displayed in a pie chart, what would the central angle of the sector representing rough conditions be? Give your answer in degrees Sea conditions
Calm Key
Moderate
Rough
A waiter can place 20 chairs at 4 tables that are the same size. How many chairs can she place at two
tables?
The total number of chairs that the waiter can place on two tables is 10 chairs.
Given, the total number chairs that the waiter can place on 4 tables is 20 chairs.
So, the number of chairs that the waiter can place on 1 chair = 20/4 chairs = 5 chairs.
Applying the concept of ratio and proportion,
Now, the total number of chairs that can be placed on 2 chairs will be =
= 2 × 5 chairs = 10 chairs
So, the total number of chairs that the waiter can place on 2 tables is 10 chairs.
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The five number summary for a set s" data is given below. Min Q1 Median Q3 Max 53 59 63 66 88 What is the interquartile range of thi set of data? Enter just the number as your answ t. For example, if you found that the interquartilerange was 22 you would enter 22 Provide your answer below:
For the given five number summary, the interquartile range is 7.
The five number summary is a set of descriptive statistics that provides a concise summary of a data set. It consists of the following five values:
Minimum: the smallest value in the data setFirst quartile (Q1): the value below which 25% of the data fallsMedian (Q2): the value that separates the data into two equal halves (i.e., 50% of the data falls above the median, and 50% falls below it)Third quartile (Q3): the value below which 75% of the data fallsMaximum: the largest value in the data setIn this problem, the five number summary is: 53 59 63 66 88
Hence,
Q1 = 59
Q3 = 66
Interquartile range = Q3 - Q1
= 66 - 59
= 7
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A quality control inspector will measure the salt content (in milligrams) in a random sample of bags of potato chips from an hour of production. Which of the following would result in the smallest margin of error in estimating the mean salt content μ? (a) 90% confidence; n=25 (b) 90% confidence; n=50 (c) 95% confidence; n=25 (d) 95% confidence; n=50 (e) n=100 at any confidence level
(d) 95% confidence; n=50 (e) n=100 at any confidence level
What is interval estimate?
Interval estimation in statistics is the use of sample data to estimate an interval of feasible values for a parameter of interest. In contrast to point estimate, which produces a single value.
The margin of error for a confidence interval estimate of the population mean decreases with an increase in the sample size and with an increase in the confidence level.
Therefore, options (b) and (d) are better than (a) and (c) because they have a larger sample size. Between options (b) and (d), a larger sample size would result in a smaller margin of error.
Therefore, option (d), which has a sample size of 50 and a higher confidence level of 95%, would result in the smallest margin of error in estimating the mean salt content μ. Option (e) with a sample size of 100 would also result in a smaller margin of error, but the confidence level is not specified, so it cannot be compared directly to the other options.
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Which of the equations are true identities?
A. (a+b)(2a+1)=a(2a+2b+1)
B. (n+2) 2 −n 2 =4(n+1)
if 5/8 of a number is 565 ,what is 5/4 of the number?
The number of 5/4 is 1130
how to find the number x?
writing this as an algebraic expression.
An algebraic expression in mathematics is an expression that is made up of variables and constants, along with algebraic operations (addition, subtraction, etc.). Expressions are made up of terms. Also, solve questions in Algebraic Expressions Worksheets
[tex]\frac{5}{8}x[/tex] = 565
[tex]\frac{5x}{8}[/tex] = 565
5x = 565 * 8
5x = 4520
x = [tex]\frac{4520}{5}[/tex]
x = 904
so the number is 904
What is 5/4 of 904 ?
[tex]\frac{5}{4}[/tex] * 904
= 1130
so the answer to this question is 1130
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Suppose the demand for a product is given by D(p) = -5p^2 + 270.000use the calculus-based definition of elasticity to find the price for the item that corresponds to an elasticity of 1.2. (give your answer to two decimal places.)Price = $ ____
The price that corresponds to an elasticity of 1.2 for this product is $59.85.
Elasticity is a fundamental concept in economics that measures the responsiveness of demand to changes in price. In this problem, we will use the calculus-based definition of elasticity to find the price that corresponds to a specific elasticity value.
The elasticity of demand measures the percentage change in quantity demanded that results from a 1% change in price. Mathematically, the elasticity of demand is defined as:
E = (%ΔQ / %ΔP) * (P / Q)
where E is the elasticity of demand, %ΔQ is the percentage change in quantity demanded, %ΔP is the percentage change in price, P is the price of the product, and Q is the quantity demanded.
To find the price that corresponds to an elasticity of 1.2, we need to solve for P in the elasticity equation above. Since we know the demand function D(p) = -5p² + 270,000, we can find Q by taking the derivative of the demand function with respect to price:
Q = D'(p) = -10p
Then, we can plug in Q and E = 1.2 into the elasticity equation and solve for P:
1.2 = (%ΔQ / %ΔP) * (P / Q)
1.2 = (%ΔQ / -1%) * (P / (-10p))
1.2 = (%ΔQ / -0.01) * (1 / -10)
%ΔQ = -0.0144
This means that a 1% increase in price will result in a 0.0144% decrease in quantity demanded, given an elasticity of 1.2. Now, we can plug in %ΔQ and Q into the demand function to solve for P:
-0.0144 = (D(p + 0.01P) - D(p)) / D(p)
-0.0144 = (-5(p + 0.01P)² + 270,000 - (-5p² + 270,000)) / (-5p² + 270,000)
-0.0144 = (-0.1P - 0.0005p²) / (-5p² + 270,000)
P = $59.85
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I require help on this math
The perimeter of the triangle FGH is 7.
What is perimeter?Perimeter is defined as space around the any two dimensional figure. It is given by the sum of all the sides of the figure.
Given the triangle FGH which is formed by connecting the midpoints of the side of the triangle CDE.
The length of the sides of the triangle are given as, CD=6, DE=4, EC=4.
To find the perimeter of the triangle FGH,
Since FGH are the midpoints of CD,DE and EC respectively.
Then, FH=1/2 DE, FG=1/2EC, GH=1/2DC.
Perimeter of a triangle = sum of the length of sides.
= 1/2(DE+EC+DC)
= 1/2(4+6+4)
Perimeter of a triangle = 7
Hence, the perimeter of the triangle FGH is 7.
Question:
Triangle FGH is formed by connecting the midpoint of the sides of the triangle CDE. The length of the sides is CD=6, EC=4, DE=5. find the perimeter of the triangle FGH.
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I need help with this question.
The measure of angle C in triangle BCD is 39⁰.
option B.
What is the measure of angle C in ΔBCD?
The measure of angle C in triangle BCD is calculated by applying the following formula as shown below.
Angle D = 92 ⁰ ( vertically opposite angles are equal )
∠CDB = 180 - 92⁰ = 88⁰ ( sum of angles on a straight line )
∠AED = 180 - (35 + 92 ) ( sum of angles in a triangle)
∠AED = 53⁰
In the quadrilateral FBDE,
angle B = 180 - 53 ( opposite angles of a cyclic quadrilateral are supplementary)
angle B = 127
∠CBD = 180 - 127 ( sum of angles on a straight line)
∠CBD = 53⁰
Then, the value of ∠BCD is calculated as;
∠BCD + ∠CBD + ∠CDB = 180 ( sum of angles in a triangle )
∠BCD + 53 + 88 = 180
∠BCD + 141 = 180
∠BCD = 180 - 141
∠BCD = 39⁰
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Question 5(Multiple Choice Worth 2 points)
(Linear Functions LC)
Which of the following equations represents a linear function?
Ox= 3
Oy=2x-5
Oy=³x²
O3x-6=4
All the linear functions are,
⇒ x = 3
⇒ y = 2x - 5
⇒ 3x - 6 = 4
What is an expression?Mathematical expression is defined as the collection of the numbers variables and functions by using operations like addition, subtraction, multiplication, and division.
We know that;
Degree of variables in a linear function is always one and the graph shows the straight line.
So, By all options;
All the linear functions are,
⇒ x = 3
⇒ y = 2x - 5
⇒ 3x - 6 = 4
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If a trend line has equation y = 15 + 0.8x, what type of association would you expect the data to have?
Alexandra is making a set of bowls in her pottery class.
She has 3/4 pound of clay.
She needs 3/2 pounds of clay to make one whole set of bowls.
Use the drop-down menus to complete each of the statements below about the bowls
that Alexandra can make.
The amount of full sets Alexandra can make would be 3.
What is meant by fraction?When there is no common factor between the fraction's numerator (top) and denominator (bottom), the fraction is said to be in its simplest form.
A fraction is a piece of the entire. In mathematics, the number is represented as a quotient, where the numerator and denominator are divided. Both are integers in a straightforward fraction. In the numerator or denominator of a complex fraction is a fraction.
A fraction is a piece of a whole number and a means to divide a number into pieces that are each equal. The numerator, also known as the number of equal parts being counted, is expressed as being greater than the denominator, also known as the number of parts in the entire.
Multiply both the numerator and denominator of 5 / 2 by 2.
5/2 × 2/2 = 10/4
The sets of bowls you can make, divide by 3 / 4.
(10/4)/(3/4) = 3 and 1/3.
The amount of full sets Alexandra can make is 3.
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Which of the following predicate logic expressions is the correct translation of the definition of the limit of a real-valued function f(x) of a real variable x at a point a in its domain? The limit of f(x) as the variable x approaches a is L if for every real number ε > 0 there exists a real number 8 >0 such that If(x) – LI<ɛ whenever 0 < lx – al
The correct translation of the definition of the limit of a real-valued function f(x) of a real variable x at a point a in its domain in predicate logic expressions.
For all ε > 0, there exists δ > 0 such that for all x, if 0 < |x-a| < δ, then |f(x)-L| < ε.
In symbols, this can be written as:
∀ε > 0, ∃δ > 0 such that ∀x, (0 < |x - a| < δ) → (|f(x) - L| < ε).
Note that "ε" and "δ" are the Greek letters epsilon and delta, respectively, and they are used to represent small positive numbers. This definition says that if we want the limit of f(x) to be L, we can find a positive number δ such that the distance between f(x) and L is less than ε whenever x is within δ units of a (but not equal to a). The limit is said to exist if we can find such a δ for any value of ε.
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Suppose an urn contains 5 blue chips and 3 red chips. Chips are drawn without replacement until either the first red chip is drawn, or until three blue chips are drawn. Let X be the number of blue chips drawn and Y the number of red chips drawn. (a) Find pX,Y , the joint pmf of X and Y . (You might want to make a table.) (b) Find the marginal pmfs pX and pY . (c) Find the conditional pmfs pX|Y (x|0) and pX|Y (x|1).
The joint pmf of X and Y 5/8, the marginal pmfs pX and pY is 3/7 and the conditional pmfs pX|Y (x|0) and pX|Y (x|1) is 5/28 and 41/56.
The probability that the chosen combination has a certain property can be set up by dividing the number of combinations where the property holds by the total number of possible combinations.
The order of the chips chosen does not matter so outcomes can be considered as combinations. We must choose 3 of the 9 chips in the box, where X are chosen from the 3 red chips, and Y are chosen from the 2 white, and the remainder are chosen from the 4 black.
For x∈{0,1,2,3}and y∈{0,1,2}, the probability that x red chips and y white chips are chosen is
To determine the marginal distribution of X, we need not distinguish white chips from black. X chips are chosen from the 3 red chips, and the remainder are chosen from the 5 non-red chips. For x∈{0,1,2,3} the probability that x chips are chosen is,
P(x) = (³ₓ) (₃-ₓ⁶) / (⁹₃)
Similarly, in determining the marginal distribution of Y, we choose Y white chips from among the 3 available, then choose the remaining chips from the 5 non-white chips.
c) To find the conditional distribution,
P(X|Y=0)= 5/28
P(X|Y=1)= 41/56
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The data below represent commute times (in minutes) and scores on a well-being survey. Complete parts (a) through (d) below.
Commute Time (min),x Well-Being Index score,y
5 69.2
15 68.4
30 67.5
35 67.3
60 66.3
84 66.1
105 64.6
(a) Find the least-squares regression line treating the commute time, x, as the explanatory variable and the index score, y, as the response variable.
(b) Interpret the slope and y-intercept, if appropriate.
(c) Predict the well-being index of a person whose commute is 35 minutes.
(d) Suppose Barbara has a 15-minute commute and scores 68.5 on the survey. Is Barbara more "well-off" than the typical individual who has a 15-minute commute?
(a) The least-squares regression line treating the commute time, x, as the explanatory variable and the index score, y, as the response variable is y = 70.22 - 0.075x
(b) The slope and y-intercept, if appropriate is the slope of the regression line, -0.075, indicates that for every additional minute of commute time, the well-being index score decreases by an average of 0.075.
(c) The well-being index of a person whose commute is 35 minutes is the slope of the regression line, -0.075, indicates that for every additional minute of commute time, the well-being index score decreases by an average of 0.075.
(d) Yes, she is. Barbara more "well-off" than the typical individual who has a 15-minute commute.
Regression of Commute Times(a) To find the least-squares regression line, we need to find the slope and y-intercept of the line that minimizes the sum of the squared vertical distances between the actual data points and the predicted values on the line. Using a calculator or software, we get:
Slope: b = -0.075
y-intercept: a = 70.22
Therefore, the least-squares regression line is:
y = 70.22 - 0.075x
(b) The slope of the regression line, -0.075, indicates that for every additional minute of commute time, the well-being index score decreases by an average of 0.075. The y-intercept, 70.22, represents the predicted well-being index score for someone with a commute time of 0 minutes (which is not a meaningful value in this context).
(c) To predict the well-being index of a person with a commute time of 35 minutes, we can substitute x = 35 into the regression equation and solve for y:
y = 70.22 - 0.075(35) = 67.97
Therefore, the predicted well-being index score for someone with a commute time of 35 minutes is 67.97.
(d) To determine whether Barbara is more "well-off" than the typical individual with a 15-minute commute, we can compare her actual well-being index score of 68.5 to the predicted score based on the regression equation:
y = 70.22 - 0.075(15) = 68.47
Barbara's score of 68.5 is slightly higher than the predicted score of 68.47, which suggests that she is somewhat better off than the typical individual with a 15-minute commute. However, we should note that this comparison is based on a single data point and may not be representative of the larger population.
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Help asap Functions hand j are inverses. When x is -10, the value of h(x) is 7, or h (-10) = 7.
a. What is the value of j(7)? Select Choice
b. Determine if each point is on the graph of h, on the graph of j, or neither.
(-10.7) Select Choice
(7.-10) Select Choice
The value of the function j(7) will be 1 / 10.
What is an inverse function?An inverse function is defined as the function in which the two variables are changed by each other and the function is solved for the value of an independent variable.
F(x) = 2x
y = 2x
The inverse function is calculated as:-
x = 2y
y = x / 2
F'(x) = x / 2
Given that Functions hand j are inverses. at the value of x = -10, the value of h(x) is 7, or h (-10) = 7.
The value of the function j(7) will be equal to 1 / 10 for the given data.
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A map of kano is drawn to a scale of 1:50000 on the map the airport covers an area of 8cm2 . find the true area of the airport in hectares (1ha = 10000m2.
The area covered by the airport in 2000000 m² and 200 Ha.
What is Scale Factor?A scale factor is a numerical value that can be used to alter the size of any geometric figure or object in relation to its original size. It is used to find the missing length, area, or volume of an enlarged or reduced figure as well as to draw the enlarged or reduced shape of any given figure. It should be remembered that the scale factor only affects how big a figure is, not how it looks.
Given:
A map of kano is drawn to a scale of 1:50000.
The area that airport covers = 8 cm²
Also, 1 ha = 10000 m²
if it's 1 cm, then real life would be 50,000 cm.
Then, 50,000 cm x 50,000 cm = 2,500,000,000 cm² in real life.
Now, the Area of Airport in real
= 8 x 2,500,000,000
= 20,000,000,000 cm²
= 2000000 m²
= 2000000 / 10000
= 200 Ha
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7.4⋅10
−8
−6.7⋅10
−9
=7, point, 4, dot, 10, start superscript, minus, 8, end superscript, minus, 6, point, 7, dot, 10, start superscript, minus, 9, end superscript, equals
The value of the equation 7.4×10⁻⁸=6.7×10⁻⁹ is 6.73×10⁻⁸
What is Number system?A number system is defined as a system of writing to express numbers.
The given expression is 7, point, 4, dot, 10, start superscript, minus, 8, end superscript, minus, 6, point, 7, dot, 10, start superscript, minus, 9, end superscript, equals
7.4×10⁻⁸=6.7×10⁻⁹
Now let us take 6.7×10⁻⁹ to left hand side
7.4×10⁻⁸-6.7×10⁻⁹
Factor 10⁻⁸ from both terms
10⁻⁸(7.4-0.67)
6.73×10⁻⁸
Hence, 6.73×10⁻⁸ is the value of the equation 7.4×10⁻⁸=6.7×10⁻⁹
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The complete question will be as below
Find the simplest form of the equation 7.4×10⁻⁸=6.7×10⁻⁹?
A kite is flying at an angle of elevation of 43o. If the string is 37 m long and is stretched tight, find the height of the kite to the nearest tenth of a meter.
The required height of the kite to the nearest tenth of a meter is 25.2 m.
What is the sine ratio in a triangle?In a right-angled triangle, the ratio of the perpendicular (with respect to an angle) to the hypotenuse gives the sine value of that angle. This relationship is used to solve problems regarding height and distance chapter.
Draw the figure as per the given instructions.
The position of the kite is at A and the string is AC which is 37 m long. The angle of elevation is ∠ACB, which is 43°. The height of the kite is AB.
So, apply the formula of sine ratio (perpendicular/hypotenuse) = sinα, where α be the opposite angle of that perpendicular.
AB/AC = sin∠ACB
AB/37 = sin43°
AB = 37sin43°
≈37×0.68
= 25.16
≈ 25.2
Therefore, the obtained answer is 25.2 m.
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determine the convergence or divergence of the sequence with the given nth term. if the sequence converges, find its limit. (if the quantity diverges, enter diverges.) an = (5√n) / (5√n + 6)
The sequence with nth term an = (5√n)/(5√n + 6) diverges.
To determine the convergence or divergence of the sequence with the given nth term, we can use the limit comparison test by comparing it with the divergent series 1/n.
We have
lim n→∞ an/(1/n) = lim n→∞ (5√n)/(5√n + 6) * n = 5/5 = 1
Since the limit is a positive finite number, and the series 1/n diverges, the series with the nth term also diverges by the limit comparison test.
Therefore, the sequence with the nth term an = (5√n)/(5√n + 6) diverges.
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What is the solution to this equation? 3v(1/8-x)=-1/2
The value of x in 3v(1/8-x)=-1/2 is 10 2/7
What are linear equations?A linear equation only has one or two variables. No variable in a linear equation is raised to a power greater than 1 or used as the denominator of a fraction.
The highest power of variable in a linear equation is 1
3v(1/8-x)=-1/2
square both sides to eliminate the root
3²v(1/8-x)² =(-1/2)²
= 9(1/8-x) = 1/4
= 9/8 - 9x = 1/4
multiply through by 8
9 - 72x = 2
72x = 9-2
72x = 7
x = 72/7
x = 10 2/7
therefore the value of x is 10 2/7
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What is the floor and ceiling of 0.561
What is the inverse of function ?
f(x) = 64x³-1
O A.
O B.
O c.
O D.
f¹ (2) = 4√x-1
f¹(x) = +¹
4
f¹(z) = Vz+1
64
f¹(x)=√x-4+1
F
The inverse of the function f(x) = 64x³ - 1 is [tex]f^{-1}(x) = (\frac{3\sqrt{(x + 1)}}{4})[/tex].
What is an inverse function?First to be an inverse function that function needs to one to one function, meaning every different preimage must correspond to a different image.
We can obtain the inverse of a function by switching the variables x and y with their respective positions and solving for y in terms of x.
Given, A function f(x) = 64x³ - 1.
Let, y = 64x³ - 1.
64x³ = y + 1.
x³ = (y + 1)/64.
[tex]x = (\frac{(y + 1)}{64})^{\frac{1}{3}[/tex].
[tex]x = (\frac{3\sqrt{(y + 1)}}{4})[/tex].
[tex]y = (\frac{3\sqrt{(x + 1)}}{4})[/tex].
[tex]f^{-1}(x) = (\frac{3\sqrt{(x + 1)}}{4})[/tex].
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equilateral triangle $abc$ has been creased and folded so that vertex $a$ now rests at $a'$ on $bc$ as shown. if $ba'
The length of PQ is (1 + √(3))/2, which is answer choice (C) equilateral triangle, angle BAC measures 60 degrees.
Since ABC is an equilateral triangle, angle BAC measures 60 degrees. Also, since A' is the midpoint of BC and A is folded onto A', angle BAA' and CAA' are both 30 degrees.
Now, let's use the Law of Cosines to find the length of PQ. Let x be the length of PQ. Then, applying the Law of Cosines to triangles ABP and AQC, we have:
BP² = x² + 1 - 2x cos30
CQ² = x² + 4 - 4x cos30
Since BP = CQ, we can set the two expressions equal to each other and simplify:
x² - 2x cos30 + 3 = 0
Solving for x using the quadratic formula, we get:
x = cos30 +/- √(cos²30 - 3)
Since cos30 = √(3)/2, we have:
x = 1/2 +/- √(3)/2
Since x must be positive, we take the positive root and get:
x = 1/2 + √(3)/2 = (1 + √(3))/2
Therefore, the length of PQ is (1 + √(3))/2, which is answer choice (C).
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The question is -
Equilateral triangle ABC has P on AB and Q on AC. The triangle is folded along PQ so that vertex A now rests at A' on side BC. If BA'=1 and A'C=2 then the length of the crease PQ is
[tex]\text{(A) } \frac{8}{5} \text{(B) } \frac{7}{20}\sqrt{21} \text{(C) } \frac{1+\sqrt{5}}{2} \text{(D) } \frac{13}{8} \text{(E) } \sqrt{3}[/tex]
A partially-filled water tank contains 300 gallons of water. water is then pumped into the tank at a constant rate. After n minutes, the amount of water, w, in the tank is W = 300 + 9 gallons. After how many minutes will W = 450 gallons?
A. 33 1/3
B. 5
C. 50
D. 16 2/3
Show your work
The amount of time when the water tank will have 450 gallons will be 16²/₃ minutes. The correct option is D.
What is an expression?Expression in maths is defined as the collection of numbers variables and functions by using signs like addition, subtraction, multiplication, and division.
Numbers (constants), variables, operations, functions, brackets, punctuation, and grouping can all be represented by mathematical symbols, which can also be used to indicate the logical syntax's order of operations and other features.
Given that a partially-filled water tank contains 300 gallons of water. water is then pumped into the tank at a constant rate. After n minutes, the amount of water, w, in the tank is W = 300 + 9n gallons.
The time will be calculated as:-
W = 300 + 9n
450 = 300 + 9n
9n = 150
n = 150 / 9
n = 16²/₃ minutes.
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If China’s GDP was proportional to that of the U.S., what would China’s expected GDP be?
The China’s expected GDP is 17lac crore.
What is a proportion?A proportion is a fraction of a total amount, and the measures are related using a rule of three.
The relations between variables, either direct or inverse proportional, can be built to find the desired measures in the problem.
Given that China’s GDP was proportional to that of the U.S
Now,
Economic growth is defined as a rise in economic activity and product availability. Real GDP has grown throughout this time period as well proportional to that of the U.S. In light of the current circumstances, the level of prices in the United States would increase, and the GDP would increase, as there is economic growth in China and Europe.
Therefore, by the given proportion the answer will be 17lac crore.
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Find 2×2_matrix which transforms the APQR having vectices P(3,2), Q(5,6) and R (4,-1) into AP'Q'R' having vertices P'(2,3), Q'(6,5) and R'(-1,4).
The transformation rule for triangle PQR into triangle P'Q'R' is given as follows:
(x,y) -> (y,x).
Which is a reflection over the line y = x.
How to obtain the transformation?The vertices of the original triangle are given as follows:
P(3,2), Q(5,6) and R (4,-1).
The vertices of the transformed triangle are given as follows:
P'(2,3), Q'(6,5) and R'(-1,4).
The x-coordinate and the y-coordinate were exchanged, hence the rule is defined as follows:
(x,y) -> (y,x).
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Ron buys a turtle tank for Tiny, his pet turtle. The tank is 16 inches tall. The base of the tank has an area of 485 square inches. What is the volume of Tiny's tank?
Answer: To find the volume of the tank, we need to determine the length and width of the base. We can use the area formula for rectangles:
Area = length * width
Since the area of the base is given as 485 square inches, we can rearrange this equation to find the length or width:
length * width = 485
Let's call the length "l". Then, the width is 485/l. The volume of the tank is the product of the area of the base and the height:
Volume = Area * height = (l * 485/l) * 16 = 485 * 16 = 7760 cubic inches.
So the volume of Tiny's tank is 7760 cubic inches.
Step-by-step explanation:
log₂ (x²-100) - log₂ (x + 10) = 1
Answer:
x = 12
Step-by-step explanation:
To find the value of x, use logarithmic rules to solve the given equation.
Given logarithmic equation:
[tex]\log_2 (x^2-100) - \log_2 (x + 10) = 1[/tex]
[tex]\textsf{Apply the log quotient law:} \quad \log_ax - \log_ay=\log_a \left(\dfrac{x}{y}\right)[/tex]
[tex]\implies \log_2 \left(\dfrac{x^2-100}{x+10}\right) = 1[/tex]
Factor the numerator x² - 100:
[tex]\implies \log_2 \left(\dfrac{(x-10)(x+10)}{x+10}\right) = 1[/tex]
Cancel the common factor (x + 10):
[tex]\implies \log_2 \left(x-10\right) = 1[/tex]
[tex]\textsf{Apply the log law:} \quad \log_ab=c \iff a^c=b[/tex]
[tex]\implies 2^1=x-10[/tex]
Simplify and solve for x:
[tex]\implies 2=x-10[/tex]
[tex]\implies 2+10=x-10+10[/tex]
[tex]\implies 12=x[/tex]
A cone has a volume of 1,230.88 units^3 and a diameter of 14 units. How many units is the height of the cone? Use 3.14 for pi.
Answer:
8 Units
Step-by-step explanation:
The volume of a cone can be calculated using the formula:
V = (1/3) * π * r^2 * h
where V is the volume, π is pi, r is the radius, and h is the height of the cone. We are given the volume and the diameter, so we can use the diameter to find the radius:
d = 2r
14 = 2r
r = 7
Now we can substitute the values for the volume and radius into the formula and solve for the height:
V = (1/3) * π * r^2 * h
1230.88 = (1/3) * 3.14 * 7^2 * h
1230.88 = (1/3) * 3.14 * 49 * h
1230.88 = (49/3) * 3.14 * h
1230.88 = 153.86 * h
h = 1230.88 / 153.86
h = 8
So, the height of the cone is approximately 8 units