The trigonometric expression 11 cos (θ − π/4) in the form a sin Bθ + b cos Bθ is 11/√2 cos θ.
What is the trigonometric expression?
The write the trigonometric expression in the required, we can start by using the second formula:
a sin Bθ + b cos Bθ = a^2 + b^2 cos(Bθ − C)
where;
C = arctan(a/b) and b > 0Let a = 11 and b = 11/√2, and Bθ = θ − π/4.
Then C = arctan(a/b) = arctan(11/(11/√2)) = arctan(√2).
Substituting these values, we get:
11 cos (θ − π/4) = 11/√2 cos (θ − π/4 + arctan(√2))
= 11/√2 cos (θ - π/4 + π/4)
= 11/√2 cos θ
Thus, we have written the trigonometric expression 11 cos (θ − π/4) in the form a sin Bθ + b cos Bθ as 11/√2 cos θ.
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Sue is playing darts. So far, she has hit the bullseye 4 times and missed the bullseye 10 times. What is the experimental probability that Sue will hit the bullseye on her next toss?
The experimental probability that Sue will hit the bullseye on her next toss is 2/7.
What is the experimental probability?
The proportion of outcomes where a specific event occurs in all trials, not in a hypothetical sample space but in a real experiment, is known as the empirical probability, relative frequency, or experimental probability of an event.
Here, we have
Given: Sue is playing darts. So far, she has hit the bullseye 4 times and missed the bullseye 10 times.
We have to find the experimental probability that Sue will hit the bullseye on her next toss.
experimental probability = 4/(4+10) = 4/14 = 2/7
The next toss = P = 2/7
Hence, the experimental probability that Sue will hit the bullseye on her next toss is 2/7.
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I need help with please as fast as possible
RT is equal to 46 and angle ∠VUT is 26°.
What is a quadrilateral?A quadrilateral is a fοur-sided pοlygοn with fοur edges and fοur cοrners in geοmetry. The name cοmes frοm the Latin wοrds quadri, a variatiοn οf fοur, and latus, which means "side."
Tο find x nοte that the diagοnals οf a rectangle are cοngruent. This means that RT = SU.
So, 3x + 8 = 6x - 7
8 + 7 = 6x - 3x
3x = 15
x = 15/3
x = 5
RT = 2RV
= 2(3x + 8)
= 2(3(5) + 8)
= 2(15+ 8)
= 2(23)
= 46
Thus, RT is equal to 46
For m∠VUT,
Given ∠VRU = 64°
Using complementary angles
64° + ∠VRS = 90°
∠VRS = 90° - 64°
∠VRS = 26°
Now, using alternate angle theory as RT is a transverse
∠VRS = ∠VTU = 26°
Now, as the diagonals are congruent ∠VTU = ∠VUT
∠VUT = 26°
Thus, RT is equal to 46 and angle ∠VUT is 26°.
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Ina and Lane share a 24-ounce bucket of clay. By the end of the week, Gina has used
3
8
of the bucket, and Lane has used
1
4
of the bucket of clay. How many ounces are left in the bucket?
9 ounces are left in the bucket. The solution has been obtained by using the arithmetic operations.
What are arithmetic operations?
It is asserted that the four basic operations, usually referred to as "arithmetic operations," can explain all real numbers. Quotient, product, sum, and difference are the next four mathematical operations after division, multiplication, addition, and subtraction.
We are given that Gina and Lane share a 24-ounce bucket of clay.
Gina has used [tex]\frac{3}{8}[/tex] of the bucket of clay.
So, we get
[tex]\frac{3}{8}[/tex] of 24 = 9 ounces
Similarly, Lane has used [tex]\frac{1}{4}[/tex] of the bucket of clay.
So, we get
[tex]\frac{1}{4}[/tex] of 24 = 6 ounces
Using the addition operation, we get
Total clay used = 9 + 6 = 15 ounces
Now, using the subtraction operation, we get
Ounces left = 24 - 15 = 9 ounces
Hence, 9 ounces are left in the bucket.
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Question: Gina and Lane share a 24-ounce bucket of clay. By the end of the week, Gina has used 3/8 of the bucket, and Lane has used 1/4 of the bucket of clay. How many ounces are left in the bucket?
find the area of the triangle
Answer:
The Area of the Triangle = 81.77
Step-by-step explanation:
Using the Formula for the Area of the Triangle given two sides and the included angle between them
Then the Area = [tex]\frac{1}{2} *13*15*sin(57)=81.77[/tex]
Hope it is Helpful
Suppose a point has polar coordinates (4, - (3pi)/4) with the angle measured in radians. Find two additional polar representations of the point. Write each coordinate in simplest form with the angle in [- 2pi, 2pi]
If a point has polar coordinates (4, - (3pi)/4) with the angle measured in radians, two additional polar representations of the points are (4, 5pi/4) and (4, -11pi/4)
To find two additional polar representations of the point (4, - (3pi)/4), we need to add and subtract integer multiples of 2pi from the angle measure while keeping the same radius.
First, let's add 2pi to the angle measure:
(4, - (3pi)/4 + 2pi) = (4, 5pi/4)
This gives us a new polar representation of the same point with a positive angle measure within the interval [-2pi, 2pi].
Next, let's subtract 2pi from the angle measure:
(4, - (3pi)/4 - 2pi) = (4, -11pi/4)
This gives us another polar representation of the same point with a negative angle measure within the interval [-2pi, 2pi].
Therefore, the three polar representations of the point (4, - (3pi)/4) are:
(4, - (3pi)/4)
(4, 5pi/4)
(4, -11pi/4)
Each of these polar representations describes the same point in the polar coordinate system, but with a different angle measure.
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Same facts as in #3, except now Elaine can set aside $50 per month. What rate of return does she need on her account? _____
The rate of return that Elaine needs on her account is of 22.16% to reach her goal of $4,000 in 4 years.
To calculate the rate of return for a savings account, you can use the formula:
Rate of return = [tex](Ending balance / Beginning balance)^{(1/n) - 1}[/tex]
where n is the number of years.
In Elaine’s case, she needs to save up $4,000 in 4 years. If she can set aside $50 per month, she will have saved $2,400 in 4 years. To calculate the rate of return she needs on her account, we can use the formula above.
To calculate the rate of return she needs on her account, we can use the formula above.
Let’s assume that she has $0 in her account at the beginning of the 4 years and $2,400 at the end of the 4 years.
Rate of return = [tex](2400 / 0)^{(1/4) - 1}[/tex]
= [tex](2400)^{(1/4) - 1}[/tex]
= 0.2216 or 22.16%
Therefore, Elaine needs a rate of return of 22.16% on her account to reach her goal of $4,000 in 4 years.
Keep in mind that this is a very high rate of return and may not be achievable with a savings account.
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Complete question is:
Elaine needs to save up $4,000 in 4 years. If she can set aside $50 per month, what rate of return does she need on her account?
Compute the expected age of a super shopper. (Round your answer to two decimal places. )
(A) The expected age μ of a super shopper 41.81
and, (B) the standard deviation σ for ages of super shoppers is 12.67
Given the data:
Age range, years 18-28 29-39 40-50 51-61 62 and over
Midpoint (x) 23 34 45 56 67
Percent of super shoppers 9% 46% 22% 11% 12%
Midpoint, x : 23 34 45 56 67
Frequency, f : 0.09_ 0.46 _ 0.22 __0.11 __ 0.12
The mean (m)= Σfx / Σf
= [(23 × 0.09) + (34 × 0.46) + (45 × 0.22) + (56 × 0.11) + (67 × 0.12)] ÷
(0.09 + 0.46 + 0.22 + 0.11 + 0.12)
= 41.81 / 1
Therefore, Mean = 41.81
Standard Deviation:
The standard deviation (or σ) is a measure of the distribution of data from the mean. A low standard deviation means the data is clustered around the mean, and a high standard deviation means the data is more scattered.
Standard deviation = Sqrt[(Σ(X² * f) / Σf) - m²)]
= ((23²× 0.09) + (34²× 0.46) + (45²× 0.22) + (56²× 0.11) + (67²× 0.12)) / 1
1908.51 - 41.81^2
= √(160.4339)
= 12.666250
= 12.67
Complete Question:
Age range, years 18-28 29-39 40-50 51-61 62 and
over Midpoint x 23 34 45 56 67
Percent of super shoppers 9% 46% 22% 11% 12%
For the 62-and-over group, use the midpoint 67 years.
(a) Compute the expected age μ of a super shopper. (Round your answer to two decimal places.)
μ = yr
(b) Compute the standard deviation σ for ages of super shoppers. (Round your answer to two decimal places.)
σ = yr
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This is 2/6 problems finish them all each is 10 points 60 total.
Li-ming received a statement on her Certificate of Deposit showing that her investment had returned $2,240
$
2
,
240
over its life. If the Certificate of Deposit pays a simple interest rate of 3.2%
3.2
%
and her initial investment was $20,000
$
20
,
000
, how long had the money
Li-ming's initial investment was $20,000, the interest rate was [tex]3.2[/tex] % (or 0.032 as a decimal), and the total interest earned was $ [tex]2,240[/tex] .
What is the deposit pays a simple interest rate?We can use the formula for simple interest to solve for the time:
Simple Interest [tex]= Principal \times Rate \times Time[/tex]
where Principal is the initial investment, Rate is the interest rate, and Time is the time period.
In this case, we know:
Principal = $[tex]20,000[/tex]
Rate = [tex]3.2[/tex] %
Simple Interest = $ [tex]2,240[/tex]
Substituting these values into the formula, we get:
$ [tex]2,240 = $20,000 \times 0.032 \times Time[/tex]
Simplifying the equation, we get:
Time = $ [tex]2,240 / ($20,000 \times 0.032)[/tex]
Time [tex]= 3.5[/tex] years (rounded to one decimal place)
Therefore, Li-ming's money was invested for 3.5 years to earn a simple interest of $ [tex]2,240[/tex] at a rate of [tex]3.2[/tex] %.
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look at the attached file
let's recall that Bearing is always a clockwise angle using the North Line, so "of X from Y", means we draw a North line at Y and a line towards X, clockwise is our angle. Check the picture below.
four minus three times a number is less than twenty-five
Answer:
the answer could be: 1,2,3,4,5,6,7. If you are not sure about the answer than substitute the number that you pick and solve as if it was a normal equation.
Step-by-step explanation:
4-(3*x)<25
example:
4-(3*1)<25
4-3<25
1<25
Can someone please help me with this?
Answer:
It’s the last one 3/5
Step-by-step explanation:
because you just simplify 18/30 with 6, witch =3/5.
18/30 divided by 6 for each and get (=) 3/5
Mei got scores of 76,80, and 78 on her last three history exams. Write an inequality to determine the score, x, she needs on the next exam so that her average is at least 82.
Mei needs to score at least 94 on her next history exam to have an average of at least 82.
What is the least score that Mei needs on the next exam so that her average is at least 82?Let's call the score that Mei needs on her next exam "x".
To find her average score, we need to add up all four scores (including the score she hasn't taken yet) and divide by 4:
(76 + 80 + 78 + x) / 4
Now, we want this expression to be greater than or equal to 82, so we can write the following inequality:
(76 + 80 + 78 + x) / 4 ≥ 82
Multiplying both sides by 4 to eliminate the fraction, we get:
76 + 80 + 78 + x ≥ 328
Simplifying the left side, we get:
234 + x ≥ 328
Subtracting 234 from both sides, we get:
x ≥ 328 - 234
x ≥ 94
Therefore, the least score neeeded on her next history exam is 94.
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There are 32 stamps.on each roll. Mark has 288 stamps. How many rolls does he have?
Answer:
9
Step-by-step explanation:
No. of rolls Mark have =
[tex] \frac{288}{32} = 9[/tex]
pls mrk me brainliest
based on an upward-sloping normal yield curve as shown, which of the following statements is correct?there is a positive maturity risk premium.inflation must be expected to increase in the future.
The answer of the given question based on the an upward-sloping normal yield curve the answer is there is a positive maturity risk premium.
What is Curve?In mathematics and geometry, curve is continuous and smooth line or path that can be described using the mathematical equations or functions. Curves can be of different shapes and sizes, like straight lines, circles, ellipses, parabolas, hyperbolas, and more complex shapes.
Based on an upward-sloping normal yield curve, the correct statement would be: there is a positive maturity risk premium.
The maturity risk premium is the additional return that investors require to hold a longer-term bond instead of a shorter-term bond. In an upward-sloping normal yield curve, longer-term bonds have higher yields than shorter-term bonds, indicating that investors are demanding a higher return to hold longer-term bonds.
This additional return, or maturity risk premium, compensates investors for the risk that interest rates may rise in the future, which would cause the value of longer-term bonds to decline more than shorter-term bonds. Therefore, a positive maturity risk premium indicates that investors expect interest rates to rise in the future.
However, the shape of the yield curve alone does not provide information on future inflation expectations. Other factors such as economic growth, monetary policy, and geopolitical events can also influence interest rates and inflation expectations.
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Is 0.403 bigger than 0.043
Yes, 0.403 is bigger than 0.043. This is because 0.403 is 10 times bigger than 0.043, since 0.403 is four hundred and three thousandths and 0.043 is forty-three thousandths.
What is bigger?It is impossible to answer this question without more context. Depending on what is being compared, one thing may be bigger than another. For example, if comparing two numbers, the larger number would be considered bigger. If comparing two physical objects, the larger object would be considered bigger.
An easy way to check is to look at the number of digits after the decimal point; 0.403 has three digits, while 0.043 has two. This indicates that 0.403 is bigger.
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You may use the following formula: Area = length x breadth Mrs Sereme wants to bake 85 scones. Calculate how many baking pns of scones will she make if each scones' dimensions are 4cm x 4 cm.
Answer: Divide the amount that she wants to bake by the dimensions
Step-by-step explanation:
Area of 1 cone =4 times 4= 16cm squared
85/16=5.3125=5 rounded to a whole number
as shown below the tank will have a height if 2 ft and a diameter of 14 ft the tank we be made of metal. if the metal costs $22 for each square foot . how much will the metal cost in total
Using the surface area, the cost value for metal is obtained as $8,707.68.
What is surface area?
The area is the area occupied by a two-dimensional flat surface. It has a square unit of measurement. The surface area of a three-dimensional object is the space taken up by its outer surface. Square units are used to measure it as well.
To find the surface area of the tank, we need to find the area of the circular top and bottom, and the area of the cylinder.
The area of a circle is given by: A = πr², where r is the radius.
Since the diameter is given as 14 feet, the radius is 7 feet.
So, the area of the top and bottom circles is: A1 = π(7²) = 153.94 square feet (rounded to two decimal places).
The circumference of the circular base is given by: C = πd, where d is the diameter.
So, the circumference is: C = π(14) = 43.98 feet.
The height of the cylinder is given as 2 feet, and the circumference of the base is 43.98 feet.
So, the area of the curved surface of the cylinder is -
A2 = C × h
A2 = 43.98 × 2
A2 = 87.96 square feet (rounded to two decimal places).
Therefore, the total surface area of the tank is -
A = 2A1 + A2
A = 2(153.94) + 87.96
A = 395.84 square feet (rounded to two decimal places).
The cost of the metal per square foot is given as $22.
So, the total cost of the metal is -
Cost = Area × Cost per square foot
Cost = 395.84 × 22
Cost = $8,707.68.
Therefore, the metal will cost $8,707.68 in total.
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please i need help ASAP it geometry!!
Answer: 12
The answer to your geometry question is 12, I hope this helps.
How is 0.136¯¯¯¯ written as a fraction in simplest form?
Enter your answer in the box.
ANSWER QUICK FOR 50 POINTS
Answer:
17/125
Step-by-step explanation:
0.136×1000÷1×1000=136/1000
136/1000=17/125
Help pls
(2.75 x 10-2) (2.5 × 108) X
Multiplying two numbers written in scientific notation can be done by multiplying their coefficients and adding their exponents, such as [tex](2.75 x 10^-2)[/tex] x [tex]10^(-2+8)[/tex] = [tex]6.875 x 10^6[/tex].
What does an expression mean?
Instead of using estimates produced at random, it is better to use rolling integer variables that can be increasing, decreasing, or blocking. They could only help one another by sharing tools, information, or solutions to issues.
In the given question,
(2.75 x [tex]10^-2[/tex]) (2.5 × [tex]10^8[/tex]) X
To multiply two numbers written in scientific notation, you can simply multiply their coefficients and add their exponents. So:
(2.75 x [tex]10^-2[/tex]) (2.5 × [tex]10^8[/tex]) = (2.75 x 2.5) x [tex]10^(-2+8)[/tex] = 6.875 x [tex]10^6[/tex]
The result is 6.875 x 10⁶, which is the sum of (2.5 x 10⁸) and (2.75 x 10⁻²).
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1. use the graph below to complete the table and answer the following questions.
a. k =
b.What does the point (1, 3) represent in the relationship?
Answer: it represent the hours it took
Step-by-step explanation:
Answer:
Answer
Step-by-step explanation:
Step-by-step explanation:
Verifying Inverse Functions In Exercises 17-20, verify thatfandgare inverse functions algebraically. 17.f(x)=4x−9,g(x)=4x+9. 18.f(x)=−3/2x−4,g(x)=−2x+8/3. 19.
f(x)= x^3/4, g(x)=
f and g are inverse functions
17. To verify that f and g are inverse functions, we need to show that f(g(x)) = x and g(f(x)) = x.
So, f(g(x)) = f(4x+9) = 4(4x+9)−9 = 16x+4−9 = 16x−5 = x.
Similarly, g(f(x)) = g(4x−9) = 4(4x−9)+9 = 16x−5+9 = 16x+4 = x.
Therefore, f and g are inverse functions.
18. To verify that f and g are inverse functions, we need to show that f(g(x)) = x and g(f(x)) = x.
So, f(g(x)) = f(−2x+8/3) = −3/2(−2x+8/3)−4 = 6x−4−4 = 6x−8 = x.
Similarly, g(f(x)) = g(−3/2x−4) = −2(−3/2x−4)+8/3 = 3x−2+8/3 = 3x+8/3 = x.
Therefore, f and g are inverse functions.
19. To verify that f and g are inverse functions, we need to show that f(g(x)) = x and g(f(x)) = x.
So, f(g(x)) = f(x3/4) = x3/44 = x3/256 = x.
Similarly, g(f(x)) = g(x4) = x4/4 = x3/4 = x.
Therefore, f and g are inverse functions.
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Two people are standing on opposite sides of a small river. One person is located at point Q, a distance of 25 meters from a bridge. The other person is standing on the southeast corner of the bridge at point P. The angle between the bridge and the line of sight from P is 72. 2 degrees. Use this information to determine the length of the bridge and the distance between the two people
Answer:
69.56 meters
Step-by-step explanation:
Let's call the distance between the two people "y" and the length of the bridge "x". Using trigonometry, we can set up two equations:
tan(72.2°) = y / 25
tan(90° - 72.2°) = y / x
We can simplify the second equation to:
tan(17.8°) = y / x
Now we have two equations with two unknowns. We can solve for one of the variables in terms of the other and substitute into the other equation to solve for the unknowns.
First, let's solve for y in the first equation:
tan(72.2°) = y / 25
y = 25 * tan(72.2°)
y ≈ 69.56 meters
Now we can substitute this value of y into the second equation:
tan(17.8°) = y / x
tan(17.8°) = 69.56 / x
x = 69.56 / tan(17.8°)
x ≈ 202.11 meters
Therefore, the length of the bridge is approximately 202.11 meters and the distance between the two people is approximately 69.56 meters.
Sol has to paint 365. 7 square feet of wall space. He wants to paint 0. 4 of the area light green. How many square feet does he want to paint light green?
Sol wants to paint 146.28 square feet of the wall light green.
The problem states that Sol has to paint 365.7 square feet of wall space, but he wants to paint only a fraction of the total area light green. The fraction he wants to paint light green is given as 0.4.
To find out how many square feet of the wall he wants to paint light green, we need to multiply the total area of the wall by the fraction of the area he wants to paint light green.
So, we can calculate the area of the wall that Sol wants to paint light green as:
Area of wall to be painted light green = 0.4 x 365.7
= 146.28 square feet
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A window has the shape of a rectangle surmounted by a regular triangle. If the perimeter of the window is p, and the base of the rectangle is x, show that in order to obtain a window of maximum area, the following relation must be satisfied X= 1/33 (6+√3) p
Plssss I need help Asap
Answer: Let's assume that the base of the rectangle is "x", and the height of the rectangle is "y". Let's also assume that the side length of the equilateral triangle is "t".
Then, we can write the perimeter of the window as:
p = x + 2y + 3t
We want to find the value of "x" that maximizes the area of the window. The area of the window can be expressed as:
A = xy + (1/2) * t * sqrt(3) * t
where the first term represents the area of the rectangle and the second term represents the area of the equilateral triangle.
To simplify the expression, we can use the perimeter equation to eliminate "y" and "t". Solving for "y", we get:
y = (1/2) * (p - x - 3t)
Solving for "t", we get:
t = (1/3) * (p - x - 2y)
Substituting these expressions into the area equation, we get:
A = x/2 * (p - x - 2y) + (1/6) * (p - x - 2y)^2 * sqrt(3)
Expanding this expression and simplifying, we get:
A = (1/12) * (p^2 - 2px + 3x^2) * sqrt(3) + (1/2) * px - (1/2) * x^2
To find the value of "x" that maximizes this expression, we can take the derivative of "A" with respect to "x" and set it equal to zero:
dA/dx = (1/12) * (6x - 2p) * sqrt(3) + (1/2) * p - x = 0
Simplifying this expression, we get:
x = (1/33) * (6 + sqrt(3)) * p
Therefore, in order to obtain a window of maximum area, the base of the rectangle should be equal to (1/33) * (6 + sqrt(3)) times the perimeter of the window.
This took a while brainliest would be appreciated (:
The length of each side of a metal cube increases at the rate of 0.025cms' when heated. Find the rate of increase in cm²s¯¹ of the total surface area of the cube, when the length of each side is 6cm.
Plssss I really need this submission due tomorrow by 6 am
Answer:
3.06 cm^2/sec
Step-by-step explanation:
Since the shape is a cube, all sides have equal lengths, 6 cm The rate of increase of 0.025cm/s [Note: I assume the unit should be cm/s, not cms'].
See the attached spreadsheet calculation. Starting at time = 0 sec, the initial area is 36 cm^2. Each second adds 0.025 cm to each side length,.
We can express this change in length as an equation:
Initial Length (6cm) + (x seconds)*(0.025 cm/s)
So for 10 second, l = 6 cm + 0.25cm or 6.25cm
--
The area is also changing with time. At 0 seconds, the area is (6cm)^2 or 36 cm^2. At 10 seconds, the area is (6.25cm)*(6.25cm) or 39.06 cm^2.
The area increased from 36 cm^2 to 39.06 cm^2 in 10 seconds. That is a change rate of ((39.06 - 36 cm^2)/(10 sec):
((39.06 - 36 cm^2)/(10 sec)
((3.06 cm^2)/(10 sec)
3.06 cm^2/sec
01.(+3) + (+8)
02.(-20) + (-13)
The answer to this question is:
01. (+11)
02. (-33)
The positive integers are added directly to the positive integers as in 01 solutions 3+8 gives 11
where else in the 02 solutions both the integers are negative and when negative integers are added to each other it is added but the sign remains negative (-20)+(-13) gives (-33).
The other formulas are:
(+) × (+) = +; (+) ÷ (+) = +
(-) × (-) = +; (-) ÷ (-) = +
(+) × (-) = -; (+) ÷ (-) = -
(-) × (+) = -; (-) ÷ (+) = -
Answer:
[tex]01.(+3) + (+8) = +11)[/tex]
[tex]02.(-20) + (-13) = (-33)[/tex]
Mark has a key ring with 10 similar keys. Three are for gym lockers, 2 are car keys, 1 is a door key, and 4 are for tool boxes. If Mark selects one key without looking, what is the probability he selects a car key or door key? ling | url
The probability that Mark selects a car key or door key is 0.3
Mark has a total of 10 keys, and he is equally likely to select any one of them. Out of these 10 keys, there are 2 car keys and 1 door key. Therefore, the probability that he selects a car key or door key is the sum of the probabilities of selecting a car key and a door key.
The probability of selecting a car key is 2/10 or 1/5, since there are 2 car keys out of 10 keys.
Similarly, the probability of selecting a door key is 1/10, since there is only one door key out of 10 keys.
Therefore, the probability of selecting a car key or door key is:
P(car key or door key) = P(car key) + P(door key)
= 1/5 + 1/10
= 3/10
= 0.3
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Which algebraic property could be used to rewrite 3x ⋅ (7y ⋅ 4) as (3x ⋅ 7y) ⋅ 4?
The algebraic property that can be used to rewrite 3x ⋅ (7y ⋅ 4) as (3x ⋅ 7y) ⋅ 4 is the associative property of multiplication.
The associative property of multiplication states that the grouping of factors in a multiplication expression does not affect the result. In other words, the order in which the multiplication operations are performed does not change the final answer. Thus, we can rearrange the multiplication expression by grouping 3x and 7y together using the associative property and then multiply the product by 4 to get the same result. This property is very useful in simplifying and solving multiplication expressions with multiple factors.
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