The system of equations according to the given instructions are x + (1/2)y = -5; 2y - 12 = x
The sum of one number and half another is negative five. Twelve less than twice the second number yields the first number.
Let x represent the first number and y represent the second number.
System:
x + (1/2)y = -5
2y - 12 = x
The system of equations that represents this situation is:
x + (1/2)y = -5
2y - 12 = x
Where x represents the first number and y represents the second number.
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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?
Answer:
60 pages
Step-by-step explanation:
64 + 56 = 120
180-120 = 60 pages
Please help! Will mark brainliest!
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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6. Construct a golden spiral that involves a golden rectangle with length 4 meters
1:1.618 (the golden ratio).
To construct a golden spiral that involves a golden rectangle with length 4 meters, first draw a 4 meter long line. Divide it into two segments, such that the longer segment has a ratio of 1:1.618 (the golden ratio) to the shorter segment.
Draw a right angle at the end of the shorter segment. This creates a square that has a side length of 4 meters. Draw an arc centered at the intersection of the two line segments with a radius equal to the shorter segment and connect the ends of the arc. This creates a golden rectangle with a length of 4 meters.
Draw a quarter circle at each corner of the golden rectangle and connect the endpoints. This creates a spiral, and it is a golden spiral if the ratio of the lengths of the two line segments is 1:1.618 (the golden ratio).
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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.
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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3/a times what equals 1
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.
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
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.
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
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.
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 ______________ .
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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(a) Show the following o-fields are equal to each other: i. B= o({(-20,x); r < R}); ii. B2 = o({(-0,2); 2 € R}); [Note either of them can be taken as the definition of the Borel sigma field on the real line.]
The following o-fields are equal to each other because they both represent the Borel sigma field on the real line:
i. B= o({(-20,x); r < R})
ii. B2 = o({(-0,2); 2 € R})
Note that the Borel sigma field on the real line is the smallest sigma field containing all the open intervals on the real line. This means that any set that can be constructed from countable unions, intersections, and complements of open intervals is in the Borel sigma field.
In the first o-field, B= o({(-20,x); r < R}), the set {(-20,x); r < R} is an open interval on the real line, so it is in the Borel sigma field. Similarly, in the second o-field, B2 = o({(-0,2); 2 € R}), the set {(-0,2); 2 € R} is also an open interval on the real line, so it is also in the Borel sigma field.
Since both o-fields contain sets that are in the Borel sigma field, they are both equal to the Borel sigma field on the real line. Therefore, B= o({(-20,x); r < R}) = B2 = o({(-0,2); 2 € R}).
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Which rational number has a different value than the others?
A. 44%
B. 22/50
C. 0.44
D. 4/9
Line f has a slope of -2 and line g has a slope of -8/4. What relationship do the lines have, based on their slopes?
A.) The lines intersect.
B.)The lines intersect to form right angles.
C.)The lines are parallel.
D.)No relationship can be determined.
The lines are parallel , based on their. So the C option is correct.
What is slope formula?
The formula to find the slope between 2 coordinates of a line is given by;
m = (y₂ - y₁)/(x₂ - x₁)
Line f has a slope of -2 and line g has a slope of -8/4. To determine the relationship between the lines based on their slopes, we can compare their slopes.
If two lines have the same slope, they are parallel. If two lines have slopes that are negative reciprocals of each other, they are perpendicular (and intersect to form right angles). Otherwise, the lines are neither parallel nor perpendicular and will intersect at some point.
The slope of line g can be simplified to -2, which is the same as the slope of line f. Therefore, the lines f and g have the same slope and are parallel (option C).
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Adina makes 53112 per year and is looking to find a new apartment rental in her city. She searched online and found an apartment for 1500 per month. The recommendation is to budget between 25% and 30% of your monthly income for rent. Can Adina afford this apartment based upon the recommended interval? Explain.
( i have to show my work )
The measures of the angles of a triangle are shown in the figure below. Solve for x.
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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A toy rocket is launched from the top of a building 128 feet tall at an initial velocity of 236 feet per second. a) Give the function that describes the height
A toy rocket is launched from the top of a building 128 feet tall at an initial velocity of 236 feet per second.
a) Give the function that describes the height of the rocket in terms of time t.
b) Determine the time at which the rocket reaches its maximum height, and the maximum height in feet.
c) For what time interval will the rocket be more than 845 feet above ground level?
d) After how many seconds will it hit the ground?
The function that describes the height of the rocket in terms of time t is h(t) = -16t² + 236t + 128
The maximum height is 962.125 feet.
The rocket reaches its maximum height at 14.75 seconds.
The rocket will hit the ground after 14.88 seconds.
a) The function that describes the height of the rocket in terms of time t is given by:
h(t) = -16t² + 236t + 128
b) To determine the time at which the rocket reaches its maximum height, we need to find the vertex of the parabola. The x-coordinate of the vertex is given by:
t = -b/2a = -236/(2*-16) = 7.375 seconds
The maximum height is then found by plugging this value of t back into the function:
h(7.375) = -16(7.375)² + 236(7.375) + 128 = 962.125 feet
c) To find the time interval during which the rocket is more than 845 feet above ground level, we need to solve the inequality:
-16t² + 236t + 128 > 845
Rearranging and factoring gives:
-16t² + 236t - 717 < 0
(4t - 59)(-4t + 12) < 0
The solutions to this inequality are t = 59/4 and t = 12/4. Therefore, the rocket will be more than 845 feet above ground level for the time interval 12/4 < t < 59/4, or 3 < t < 14.75 seconds.
d) To find when the rocket will hit the ground, we need to solve the equation:
-16t² + 236t + 128 = 0
Using the quadratic formula gives:
t = (-236 ± √(236² - 4(-16)(128)))/(2(-16))
t = (-236 ± 244.28)/(-32)
The two solutions are t = 14.88 seconds and t = -0.51 seconds. Since time cannot be negative, the rocket will hit the ground after 14.88 seconds.
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A college entrance exam has a mean of 500 and a standard deviation of 100 in the population. A small liberal arts college has 100 applicants with a mean of 510. The college wants to determine whether its applicants have significantly higher scores than the general population of applicants. The data analyst obtains a t statistic of 1. 70 and a one-tailed probability of. 4. What would the probability be if the analyst performed a two-tailed test?
The probability when analyst performed a two-tailed test for the given one tailed probability is equal to 0.2.
Mean = 500
Standard deviation = 100
Number of applicants in art college = 100
Mean of the art college = 510
t-statistic = 1.70
One tailed probability = 0.4
Analyst performed a two-tailed test,
⇒ Probability value = 2(one-tailed probability value)
In a two-tailed test, deviations from the mean in both directions.
Probability of observing values are both higher and lower than the mean.
⇒ Probability for two-tailed test = Probability value for one-tailed test / 2
⇒Probability for two-tailed test = 0.4 / 2
⇒Probability for two-tailed test = 0.2
Therefore, the probability for a two-tailed test is equal to 0.2.
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Which piecewise function represents the graph?
Answer:
A
Step-by-step explanation:
trust
I’ve tried it so many times and can’t get the answer
Answer:
(364.0, 236.4)
Step-by-step explanation:
You want the vector 434∠33° written as an ordered pair.
Ordered pairThe "ordered pair" representation of a vector can take different forms. we're not sure the form your curriculum materials are asking for, so we'll show it a couple of ways.
(magnitude; direction) = (434; 33°) or maybe (434, 33)
(x, y) = 434·(cos(33°), sin(33°) ≈ (364.0, 236.4)
__
Additional comment
We like the form used in the problem statement above (434∠33°) for vectors expressed in polar form. For vectors expressed in rectangular form, many calculators work nicely with a complex number representation (364 +236.4i).
The ordered pair using a semicolon (;) separator is written that way to distinguish it from the (x, y) ordered pair. That form is used by some geometry software. A calculator may use [434, 33] or [364, 236.4] (for example), leaving it up to the user to remember what the pair represents.
Some authors use a combination of complex and polar representations like this:
A(cos(B) +i·sin(B)) = A·cis(B)
In this, the "cis" is a reminder of the "cosine + i·sine" complex number representation. It is also a longer way to write A∠B.
The second attachment shows our preferred calculator using our preferred vector notations. YMMV
→ Be sure your calculator is in "degrees" mode.
Evaluate each logarithm by using properties of logarithms and the following facts.
loga(x) = 3.1 loga(y) = 5.8 loga(z) = 1.1
(a) loga (xy)
(b) loga (x/z)
(c) loga (y7)
(d) loga (square root y)
Please explain how to figure these out. Thank you ! Any help is appreciated!
We can use the properties of logarithms to evaluate each expression. The properties of logarithms include:
1) loga (xy) = loga (x) + loga (y)
2) loga (x/z) = loga (x) - loga (z)
3) loga (x^p) = p loga (x)
4) loga (square root x) = (1/2) loga (x)
Using these properties, we can evaluate each expression as follows:
(a) loga (xy) = loga (x) + loga (y) = 3.1 + 5.8 = 8.9
(b) loga (x/z) = loga (x) - loga (z) = 3.1 - 1.1 = 2
(c) loga (y^7) = 7 loga (y) = 7 * 5.8 = 40.6
(d) loga (square root y) = (1/2) loga (y) = (1/2) * 5.8 = 2.9
Therefore, the answers are:
(a) loga (xy) = 8.9
(b) loga (x/z) = 2
(c) loga (y^7) = 40.6
(d) loga (square root y) = 2.9
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an international company had 19700 employes in one country that is 22.9 percent of there employs, how many employies do they have in all
Rounded to the nearest whole number, the international company has approximately 86,026.2 employees in total.
What formula is used to calculate a percentage?We must first divide the value by the whole sum, and then multiply the result by 100 to get the percentage.
Let's use algebra to solve the problem.
If 19,700 employees represent 22.9% of the company's total number of employees, we can write:
0.229 = 19,700 / Total number of employees
To find the total number of employees, we can isolate the variable on one side of the equation:
Total number of employees = 19,700 / 0.229
Using a calculator, we get:
Total number of employees = 86,026.2
Rounded to the nearest whole number, the international company has approximately 86,026.2 employees in total.
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Perform the following conversions. With math explanation
300 ft = m
130 cm = in
0.5hp = W
350psi = Pa
10 ????????/????????3 = ????????/m3
350 W = ????T????/ℎ
10,000 cal = ????
a)91.44 m
b)51.18 in
c)372.85 W
d)2383186.955 Pa
e)10000 ????????/m3
f)0.4667 T????/ℎ
g)41840 J
300 ft = m
1 ft = 0.3048 m, so 300 ft = 300 * 0.3048 m = 91.44 m
130 cm = in
1 in = 2.54 cm, so 130 cm = 130/2.54 in = 51.18 in
0.5hp = W
1 hp = 745.7 W, so 0.5 hp = 0.5 * 745.7 W = 372.85 W
350psi = Pa
1 psi = 6894.757 Pa, so 350 psi = 350 * 6894.757 Pa = 2383186.955 Pa
10 ????????/????????3 = ????????/m3
1 ????????/????????3 = 1000 ????????/m3, so 10 ????????/????????3 = 10 * 1000 ????????/m3 = 10000 ????????/m3
350 W = ????T????/ℎ
1 W = 0.00134 T????/ℎ, so 350 W = 350 * 0.00134 T????/ℎ = 0.4667 T????/ℎ
10,000 cal = ????
1 cal = 4.184 J, so 10,000 cal = 10,000 * 4.184 J = 41840 J
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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.
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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Find the measurements of X
pt. 2
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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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?
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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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)
What are the solutions for the equation 6x²-11x-7=0
The solutions for the equation 6x²-11x-7=0 are x = -1/2 and x = 7/3
How to determine the solutionIt is important to note that quadratic equations are defined as equations having the highest degree of x as 2.
From the information given, we have that the quadratic equation is given as;
6x²-11x-7=0
Now, multiply the coefficient of x squared with the constant value, then find the pair factors of the product that adds up to given the coefficient of x = -11
We have;
6x² - 14x + 3x - 7 =0
Group the expression in pairs
(6x² - 14x) + (3x - 7) = 0
Factor the expressions
2x(3x - 7) + 1(3x - 7) = 0
Then, we have;
(2x + 1) and (3x - 7) = 0
x = -1/2
x = 7/3
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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
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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The math coach is delivering boxes of books to classrooms. She has 495 books to deliver. Each box contains 15 books. She has already delivered 4 boxes. Write an equation, where b is the number of boxes of books, that the math coach still needs to deliver.
Answer:
The math coach has delivered 4 boxes of books, so the number of boxes she still needs to deliver is b. Each box contains 15 books, so the total number of books she still needs to deliver is 15b. She has 495 books to deliver in total. Therefore, we can write the equation:
15b = 495 - 4(15)
Simplifying the right-hand side:
15b = 495 - 60
15b = 435
Dividing both sides by 15:
b = 29
So the math coach still needs to deliver 29 boxes of books.
The number of blueberry muffins that a baker makes each day is 30% of the total number of muffins she makes.
A. On Monday, the baker makes 42 blueberry muffins. What is the total number of muffins that the baker makes on Monday?
B. On Tuesday, the baker makes a total of 60 muffins. How many blueberry muffins does the baker make on Tuesday?
The baker makes a total of 140 muffins on Monday.
The baker makes 18 blueberry muffins on Tuesday.
What are percentages?
Percentages are a way of expressing a fraction or proportion out of 100. The term "percent" literally means "per hundred." Percentages are commonly used in many areas of everyday life, including finance, science, and statistics.
To express a number as a percentage, it is multiplied by 100. For example, 0.75 is equivalent to 75 percent, since 0.75 x 100 = 75. Similarly, 50 percent is equivalent to 0.5 or 1/2.
Percentages are commonly used to express changes and comparisons. For example, if a product's price increases from $100 to $120, this represents a 20 percent increase in price. If a person scored 80 out of 100 on a test, this represents an 80 percent score.
Percentages are also used to calculate percentages of a total. For example, if a person earned $40,000 out of a total company income of $500,000, their income represents 8 percent of the total income. Similarly, if a person wants to calculate a 20 percent tip on a $50 meal, they would calculate 20 percent of $50, which is $10.
In summary, percentages are a way of expressing a fraction or proportion out of 100, and are commonly used to express changes, comparisons, and percentages of a total in everyday life.
A. Let's assume the total number of muffins the baker makes on Monday to be x.
According to the problem, the number of blueberry muffins that the baker makes is 30% of the total number of muffins.
So, 30% of x is equal to 42:
0.30x = 42
To solve for x, we can divide both sides by 0.30:
x = 140
Therefore, the baker makes a total of 140 muffins on Monday.
B. Let b be the number of blueberry muffins the baker makes on Tuesday.
According to the problem, the number of blueberry muffins is still 30% of the total number of muffins. So, we can set up the equation:
0.30(60) = b
Simplifying the left side:
18 = b
Therefore, the baker makes 18 blueberry muffins on Tuesday.
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A grocer wants to create a 40 pound mixture of almonds and cashews that will sell for $12.50 per pound. The almonds sell for $10 per pound and the cashews sell for $14 per pound. How many pounds of each should the grocer use?
The grocer should use 15 pounds of almonds and 25 pounds of cashews to create a 40 pound mixture that will sell for $12.50 per pound.
To create a 40 pound mixture of almonds and cashews that will sell for $12.50 per pound, the grocer should use a combination of both almonds and cashews that will give him the desired price. We can use a system of equations to solve for the amount of each ingredient the grocer should use. Let x be the amount of almonds and y be the amount of cashews.
The first equation is the total weight equation: x + y = 40The second equation is the total cost equation: 10x + 14y = 12.50(40)Solving for x in the first equation gives us: x = 40 - y
Substituting this value of x into the second equation gives us: 10(40 - y) + 14y = 12.50(40)
Simplifying and solving for y gives us: 4y = 100 -> y = 25
Substituting this value of y back into the first equation gives us: x + 25 = 40 -> x = 15
Therefore, the grocer should use 15 pounds of almonds and 25 pounds of cashews to create a 40 pound mixture that will sell for $12.50 per pound.
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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)
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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