What is the Measure Of arc WX in the diagram below?
Answer:
The correct option is A.
Step-by-step explanation:
Given information: Length of Arc TR is 116° and the measure of ∠WSX is 45°.
According to the angle of Intersecting Secants Theorem,
[tex]\text{Angle of Intersecting Secants}=\frac{1}{2}(\text{Major arc - Minor arc})[/tex]
Using this property we get
[tex]45^{\circ}=\frac{1}{2}(Arc(TR)-Arc(WX))[/tex]
[tex]45^{\circ}=\frac{1}{2}(116^{\circ}-Arc(WX))[/tex]
Multiply 2 on both the sides.
[tex]90^{\circ}=116^{\circ}-Arc(WX)[/tex]
[tex]Arc(WX)=116^{\circ}-90^{\circ}[/tex]
[tex]Arc(WX)=26^{\circ}[/tex]
The measure of arc WX is 26°. Therefore the correct option is A.
Which expression shows the result of applying the distributive property to 12(5y+4) ?
A. 17y + 16
B. 5y + 48
C. 60y + 48
D. 60y + 4
Select all statements below which are true for all invertible n×nn×n matrices aa and bb
a. (ab)−1=a−1b−1(ab)−1=a−1b−1
b. aba−1=baba−1=b
c. (in+a)(in+a−1)=2in+a+a−1(in+a)(in+a−1)=2in+a+a−1
d. a2b7a2b7 is invertible
e. a+ba+b is invertible f. (a+b)(a−b)=a2−b2(a+b)(a−b)=a2−b2
Answer:
The statements that are correct are:
c) [tex](in+a)(in+a^{-1})=2in+a+a^{-1}[/tex]
d) [tex]a^2b^7[/tex] is invertible.
e) [tex]a+b[/tex] is invertible.
Step-by-step explanation:
We are given that:
a and b are invertible n×n matrices.
We have to tell which of the following statements are true.
a)
[tex](ab)^{-1}=a^{-1}b^{-1}[/tex]
This statement is false.
Since:
[tex](ab)^{-1}=b^{-1}a^{-1}[/tex] and it may not be equal to the term [tex]a^{-1}b^{-1}[/tex]
b)
[tex]aba{-1}=b[/tex]
This expression could also be written as:
[tex]ab=ba[/tex]
Since on Post multiplying by a on both the sides.
But here we don't know whether the matrices are commutative or not.
Hence, the statement is false.
c)
[tex](in+a)(in+a^{-1})=2in+a+a^{-1}[/tex]
This statement is true.
since,
[tex](in+a)(in+a^{-1})=in(in+a^{-1})+a(in+a^{-1})\\\\=in^2+in.a^{-1}+a.in+aa^{-1}\\\\=in+a^{-1}+a+in\\\\=in+a^{-1}+a[/tex]
where in denote the identity matrix.
and we know that:
[tex]in^2=in[/tex]
d)
[tex]a^2b^7[/tex] is invertible.
This statement is true.
Since we know that prodct of invertible matrices is also invertible.
As [tex]a[/tex] is invertible so is [tex]a^2[/tex].
Also [tex]b[/tex] is invertible so is [tex]b^7[/tex].
Hence Product of [tex]a^2[/tex] and [tex]b^7[/tex] is also invertible.
i.e. [tex]a^2b^7[/tex] is invertible.
e)
[tex]a+b[/tex] is invertible.
This statement is true as sum of two invertible matrices is invertible.
f)
[tex](a+b).(a-b)=a^2-b^2[/tex]
This statement is false.
Since,
[tex](a+b).(a-b)=a(a-b)+b(a-b)\\\\=a.a-a.b+b.a-b.b\\\\=a^2-ab+ba-b^2[/tex]
Now as we are not given that:
[tex]ab=ba[/tex]
Hence, we could not say that:
[tex](a+b).(a-b)=a^2-b^2[/tex]
The correct statements which are true for all invertible [tex]\left({n\times n}\right)\cdot\left({n\times n}\right)[/tex] are:
(c). [tex]\left({in+a}\right)\left({in+{a^{-1}}}\right)=2in+a+{a^{-1}}[/tex]
(d). [tex]{a^2}{b^7}[/tex] is invertible.
(e). [tex]a+b[/tex] is invertible.
Further Explanation:
Given:
The matrix [tex]a[/tex] and [tex]b[/tex] are invertible [tex]\left({n\times n}\right)[/tex] matrices.
Calculation:
(a)
The statement is [tex]{\left({ab}\right)^{-1}}={a^{-1}}{b^{-1}}[/tex] false.
[tex]\begin{aligned}{\left({ab}\right)^{-1}}&={a^{-1}}{b^{-1}}\\\left({ab}\right){\left({ab}\right)^{-1}}&=\left({ab}\right){a^{-1}}{b^{-1}}\\I&\ne ab{a^{-1}}{b^{-1}}\\\end{aligned}[/tex]
The statement is [tex]{\left({ab}\right)^{-1}}={a^{-1}}{b^{-1}}[/tex] false.
(b)
The statement is [tex]ab{a^{-1}}=b[/tex].
Now multiply by a both the side.
[tex]\begin{aligned}ab{a^{-1}}a&=ba\\ab&\ne ba\\\end{aligned}[/tex]
The statement is [tex]ab{a^{-1}}=b[/tex] is false.
(c)
The statement is [tex]\left({in+a}\right)\left({in+{a^{-1}}}\right)=2in+a+{a^{-1}}[/tex].
Solve the above equation to check whether it is invertible.
[tex]\begin{aligned}\left({in+a}\right)\left({in+{a^{-1}}}\right)&=in\left({in+{a^{-1}}}\right)+a\left({in+{a^{-1}}}\right)\\&=i{n^2}+in\cdot{a^{-1}}+a\cdot in+a{a^{-1}}\\&=in+{a^{-1}}+a+in\\&=in+{a^{-1}}+a\\\end{aligned}[/tex]
The statement is true.
(d)
The statement is [tex]{a^2}{b^7}[/tex].
The product of invertible matrices is always invertible.
As [tex]a[/tex] is invertible so [tex]{a^2}[/tex] is also invertible.
As [tex]b[/tex] is invertible so [tex]{b^7}[/tex] is also invertible.
Hence, the product of [tex]{a^2}[/tex] and [tex]{b^7}[/tex] is also invertible.
The statement is true.
(e)
The statement [tex]a+b[/tex] is true as [tex]a+b[/tex] is always invertible.
(f)
The statement is [tex]\left({a+b}\right)\cdot\left({a-b}\right)={a^2}-{b^2}[/tex].
Solve the equation to check the inevitability.
[tex]\left({a+b}\right)\times\left({a-b}\right)={a^2}-ab+ba-{b^2}[/tex]
The statement is not true as [tex]ab=ba[/tex].
Hence, the correct statements which are true for all invertible [tex]\left({n\times n}\right)\cdot\left({n\times n}\right)[/tex] are:
(c). [tex]\left({in+a}\right)\left({in+{a^{-1}}}\right)=2in+a+{a^{-1}}[/tex]
(d). [tex]{a^2}{b^7}[/tex] is invertible.
(e). [tex]a+b[/tex] is invertible.
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Answer details:
Grade: High School
Subject: Mathematics
Chapter: Linear equation
Keywords: Invertible, matrices, matrix, statement, function, true, determinants, elements, inverse.
Can someone help me with 2 and 4
the ratio of girls to boys in the math club was 3.5. if there were 120 children altogether, how many girls were there?
3 girls to 5 boys
3+5 =8
120/8 = 15
3 *15 = 45 girls
HELP QUICK!!! 3. There is a special sale at the pet store. You can buy a fish bowl and get 6 free goldfish. Carlos and Olivia each buy a bowl and get 6 fish. They want to keep track of their fishes’ growth, so they carefully measure each fish and record these lengths:
1 ¼ in
1 7/8in.
1 1/4in.
1 3/4in.
1 3/8 in.
1 7/8 in.
1 3/8 in.
1 1/4in.
1 4/8in.
1 6/8in.
1 7/8in.
1 3/4in.
(a) Plot the lengths of each of the 12 goldfish on the line plot.
(b) Which is greater: the amount of fish that measure less than 1 1/2 inches in length or the amount of fish with a length greater than 1 1/2 inches?
for a, make a dot on the line plot at each of the given lengths
b) there are more fishes greater than 1 1/2
Pascal wants to find the sum of the measures of the interior angles of a polygon that has n sides. How could he do this?
Answer with explanation:
Smallest Polygon = Triangle , which has 3 sides
Sum of angles of triangle = 180°→→, Using the angle sum property of triangle.
We can find the Sum of angles of triangle, using the formula
= (n-2)×180°
= (3-2)×180°
= 1 × 180°
= 180°
A Quadrilateral has four sides.
Sum of angles of Quadrilateral = 360°
Or , using the formula
=(4-2)×180°
=2 × 180°
=360°
For, n sided Polygon
Sum of it's Interior angles = (n-2) × 180°
To prove this, we should divide that polygon into triangles, and multiply number of triangles by angle of 180°.
For, example, Quadrilateral can be divided into two triangles , sum of interior angles of Quadrilateral =2 × 180°=360°
Bob has a dog who weighs 12 pounds. His cat weighs 2/3 as much as the dog. How many pounds does his cat weigh?
In 1943 the temperature in rapid city South Dakota fell about 45 degrees in 5 minutes what was the average drop in temperature per minute
Answer:-9 degrees
Step-by-step explanation:The average drop in temperature per minute is 9 degrees.
A grocery store's receipts show that sunday customer purchases have a skewed distribution with a mean of $3030 and a standard deviation of $2121. suppose the store had 304304 customers this sunday. a) estimate the probability that the store's revenues were at least $9 comma 6009,600. b) if, on a typical sunday, the store serves 304304 customers, how much does the store take in on the worst 11% of such days?
a) The probability is approximately 0.9973.
b) The store takes in at most [tex]$\$ 15,600.00$[/tex].
a) To estimate the probability that the store's revenues were at least [tex]$\$9,500$[/tex], we need to find the z-score corresponding to this revenue value and then find the probability associated with that z-score in the standard normal distribution.
First, we calculate the z-score:
[tex]\[ z = \frac{X - \mu}{\sigma} = \frac{9500 - 3200}{2000} = \frac{6300}{2000} = 3.15 \][/tex]
Using a standard normal distribution table or a calculator, we find that the probability corresponding to [tex]$z = 3.15$[/tex] is approximately 0.9987.
So, the estimated probability that the store's revenues were at least [tex]$\$9,500$[/tex] is approximately 0.9987.
b) To find out how much the store takes in on the worst [tex]$10\%$[/tex] of such days, we need to find the revenue value corresponding to the [tex]$90\%$[/tex] percentile of the distribution.
Using the standard normal distribution table or a calculator, we find that the [tex]$90\%$[/tex] percentile corresponds to a z-score of approximately 1.28.
Now, we can find the revenue value:
[tex]\[ X = \mu + z \times \sigma = 3200 + 1.28 \times 2000 = 3200 + 2560 = 5760 \][/tex]
So, on the worst [tex]$10\%$[/tex] of such days, the store takes in at most [tex]$\$5,760.00$[/tex].
The correct question is:
A grocery store's receipts show that Sunday customer purchases have a skewed distribution with a mean of [tex]$\$ 32$[/tex] and a standard deviation of [tex]$\$ 20$[/tex]. Suppose the store had 294 customers this Sunday.
a) Estimate the probability that the store's revenues were at least [tex]$\$ 9,500$[/tex].
b) If, on a typical Sunday, the store serves 294 customers, how much does the store take in on the worst [tex]$10 \%$[/tex] of such days?
a) The probability is [tex]$\square$[/tex].
b) The store takes in at most [tex]$\$ \square$[/tex].
Gina works at a diner. Earns $6.00 hour + $43.00 tips how much does she make all together?
Heather purchased an iPad mini on sale 20% off for 263.20 what's the original price of the iPad mini
Rearrange the formula A = \pi r ^ ( 2) for r
Which inequality models this problem?
Esin started a T-shirt-printing business. She spent $2500 to purchase supplies to get started and she uses about $4.50 worth of supplies per T-shirt printed. Esin charges $25 for each T-shirt. Let s represent the number of T-shirts.
What is the minimum number of T-shirts Esin will need to sell to make a profit?
25s<2500+4.5s
On a recent business trip, a salesman traveled 52 miles the first hour, 64 miles the second hour and 49 miles the third hour.
How many miles did he average per hour?
Answer: He traveled 54.28 miles per hour as an average.
Step-by-step explanation:
Since we have given that
Number of miles traveled in first hour = 52 miles
Number of miles traveled in second hour = 64 miles
Number of miles traveled in third hour = 49 miles
Average per hour is given by
[tex]=\dfrac{3}{\dfrac{1}{x}+\dfrac{1}{y}+\dfrac{1}{z}}\\\\\\=\dfrac{3}{\dfrac{1}{52}+\dfrac{1}{64}+\dfrac{1}{49}}\\\\\\=\dfrac{3}{\dfrac{784+637+832}{40768}}\\\\\\=\dfrac{3}{\dfrac{2253}{40768}}\\\\\\=\dfrac{40768\times 3}{2253}\\\\\\=\dfrac{122304}{2253}\\\\\=54.28\ miles/hr[/tex]
Hence, he traveled 54.28 miles per hour as an average.
Nine fewer than half a number is a five more than a four times the number.define a varible, write an equation, and slove to find the number
0.5x-9 = 4x+5
add 9 to both sides
0.5x = 4x+14
subtract 4x from each side
-3.5x =14
x = 14/*3.5
x = -4
A rectangular table is 4 times as long as it is wide. If the area is 100ft^2, find the width and length of the table.
Barbara sells iced tea for $1.49 per bottle and water for $1.25 per bottle. She wrote an equation to find the number of drinks she needs to sell to earn $100.
1.25x + 1.49 = 100
What error did Barbara make in writing the equation?
A swimming pool is being filled ata rate of 15 pints per minute. How many gallons per hour is this
1 gallon = 8 pints
15/8 = 1.875 gallons per minute
60 minutes per hour
1.875 * 60 = 112.5 gallons per hour
A game of chance involves rolling a 14-sided die once. If a number from 1 to 3 comes up, you win 2 dollars. If the number 4 or 5 comes up, you win 9 dollars. If any other number comes up, you lose. If it costs 5 dollars to play, what is your expected net winnings?
The expected net winnings are -$36. That is you lose $36 according to chance.
We have given
A game of chance involves rolling a 14-sided die once. If a number from 1 to 3 comes up, you win 2 dollars.
We have to determine expected net winnings
What is the expected net winnings?The expected value is what the player can expect to win or lose if they were to play many times with the same game.
let's say that you have a 3/13 chance of winning $3 and a 2/13 chance of winning $10 and an 8/13 chance of winning $0.
Therefore if the dice is rolled 13 times then the expected to win 3X$3 + 2X$10 = $29
But you pay $5 per roll so 13 rolls would cost me 13X$5 = $65
Therefore the expected net winnings are $29-$65 = -$36
Therefore the expected net winnings are -$36.
So you lose $36 according to chance.
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A bag contains 450 blue and yellow marbles. The probability of drawing a yellow marble is 46%. How many blue marbles are in the bag?
the sum of two consecutive numbers is 73. what are the numbers?
How many 1/2 cups servings of r ice are in 3/8 cups of rice? Explain your answer
"isotherms are drawn at regular intervals; on this map, the interval between successive isotherms is ______ fahrenheit degrees."
An isotherm on a geographic map is a type of equal temperature at a given date or time. And if we talk about thermodynamics, it is a curve on a Pressure – Volume diagram at a constant temperature.
Basing from the graph, we can actually see that the interval is:
10 Fahrenheit degrees
You take a trip by air that involves three independent flights. if there is an 76% chance each specific leg of the trip is on time, what is the probability all three flights arrive on time?
The mean height of a group of students is 65.4 inches. A student joins the group who is 73 inches tall. How does this student’s height affect the mean?
A. The new mean height will still be 64.5 inches.
B. The new mean height will be greater than 65.4 inches.
C. The new mean height will be less than 65.4 inches.
D. The new mean height will be 73 inches.
Answer:
The new mean height will be greater than 65.4 inches.
Step-by-step explanation:
The mean height of a group of students is 65.4 inches. A student joins the group who is 73 inches tall.
Mean is 65.4 inches
A student joins the group who is 73 inches tall.
73 is greater than the means 64.4 inches
So mean will increase definitely
Mean will vary depends on the total number of students
The new mean height will be greater than 65.4 inches.
Describe the graph of the basic rational function, f(x)=1/x. Talk about its domain and range and compare the function to other ones, i.e. linear or quadratic or exponential.
Final answer:
The graph of f(x) = 1/x is a hyperbola with vertical and horizontal asymptotes at x = 0 and y = 0, respectively. The domain is all real numbers except 0, and the range is the same. Unlike linear, quadratic, or exponential functions, the hyperbolic shape of 1/x presents its unique characteristics needing careful domain consideration.
Explanation:
The graph of the basic rational function f(x) = 1/x is characterized by two distinct branches, one in the first quadrant (where x and y are both positive) and the other in the third quadrant (where x and y are both negative). This function is a hyperbola, and as the value of x approaches zero from either the positive or negative side, the value of f(x) increases without bound, referred to as a vertical asymptote at x = 0. Additionally, as the value of x becomes very large or very small (approaching positive or negative infinity), the value of f(x) approaches zero, creating a horizontal asymptote at y = 0. Importantly, this function is undefined at x = 0, so the domain is all real numbers except 0, (domain: ℝ \ {0}). The function takes on all real number values for the output, so the range is also all real numbers except 0 (range: ℝ \ {0}).
Comparing to other types of functions, unlike linear functions like f(x) = 7x which produces a straight line or quadratic functions such as f(x) = 2x² which graphs as a parabola, the rational function f(x) = 1/x leads to a non-linear graph with two separate branches. Exponential functions like f(x) = eⁿ¹, however, grow or decline at an increasing rate, which is a different behavior compared to the hyperbolic decline or rise of a rational function. It's vital to note that as with the function |√x|, there are points along the graph of 1/x where the function is not defined, highlighting the importance of considering the domain in function analysis.
The function 1/x² also represents a rational function but with a different characteristic as it exhibits positive blowup on both sides. This means that as x approaches zero, whether from the positive or negative side, the function tends to positive infinity, a behavior different from the reciprocal function 1/x.
Explain in a minimum of 2 sentences how to graph the equation of the absolute value function given a vertex of (-1,3) and a value of “a” equal to ½.
To graph an absolute value function with a given vertex and 'a' value, first plot the vertex. Then use the 'a' value to find and plot other points. Connect the points to create the graph.
Explanation:To graph the equation of the absolute value function given a vertex of (-1,3) and a value of “a” equal to ½, you first plot the vertex on the graph. This point will be the lowest (if 'a' is positive) or highest (if 'a' is negative) point on the graph. The vertex of the absolute value function is the point of reflection. Next, create a table to find extra points on the graph. Because 'a' is ½, count over 1 and up (or down depending on if 'a' is positive or negative) half the distance. Similarly, count over 2 and up (or down) the distance to find more points. Once you've plotted these points, you'll see the absolute value function starting to take shape. Connect the dots to create the two arms of the absolute value function that point upwards or downwards depending on the value of 'a'. This will result in a 'V' shaped graph if 'a' is positive or an upside-down 'V' shape if 'a' is negative.
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Please help! (Picture included)
Which value of b will cause the quadratic equation x2 + bx + 5 = 0 to have two real number solutions?
The Correct Answer is A:-5 :)