If David spends $10, he may call for 50 minutes.
As per the data given in the above question are as bellow,
Let x represent the duration of David's phone calls.
The following linear equation can be used to express the call's cost:
Price = x + (x - 3) (0.10)
By setting the equation's value to 10 and finding the value of x, we can determine how long David will talk on the phone if he spends $10:
10 = 0.25x + (x-3)(0.10) (0.10)
The right side of the equation is first made simpler:
10 = 0.25x + 0.10x - 0.30
Next, we may combine related words as follows: 10 = 0.35x - 0.30
10 plus 0.30 equals 0.35x when applied to both sides.
Consequently, x = 50. David talks on the phone for 50 minutes.
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help Choose Yes or No to tell if the operation will give an equivalent ratio for 8/
20
.
.
a) Does not give an equivalent ratio
b) Does not give an equivalent ratio
c) Does not give an equivalent ratio
d) Gives an equivalent ratio
What is the equivalent ratio?An equivalent ratio is a ratio that is equal in value to a given ratio, but may have a different form. Two ratios are equivalent if they have the same relationship between the terms, even if the numbers are different.
In this case, we have the ratio 8/20. The process of addition or subtraction can not be able to give us the equivalent ratio that we are looking for but the multiplication would do that. Recall that the original fraction is 8/20 so;
a) 10/22 = 5/11
b) 6/18 = 3/9
c) 4/10 = 2/5
d) 16/40 = 8/20
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In triangle HPK, k = 8, p = 14. 9 and the measure of angle H is 51 degrees. Find the area of the triangle
The area of the triangle is 59.6[tex]m^2[/tex]
Now, According to the question:
The area of a triangle is defined as the total region that is enclosed by the three sides of any particular triangle. Basically, it is equal to half of the base times height, i.e. A = 1/2 × b × h.
In the question, Area unit is not mentioned.
But i take the unit in meter.
Now, We have the given data is:
In triangle HPK:
K = 8 m
P = 14.9 m
To find the area of triangle.
We know that :
Area of Triangle is = 1/2 × b × h.
A = 1/2 × 8 × 14.9
A = 59.6 [tex]m^2[/tex]
Hence, The area of the triangle is 59.6
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PLEASE HELP I WILL GIVE BRAINLIEST
For the set of triangle vertices, find the coordinates of the vertices of the image after a dilation by the given scale
factor and center of dilation.
D(2,-2), E(4, -2), F(1, -4)
k = 1.5,, E(4,-2), centered at E
Answer:
Step-by-step explanation:I D K
5
Ty earns $14 an hour working on weekdays and
$21 an hour working on weekends. If he wants to
make at least $600 by working a total of 36 hours
in a week, to the nearest hour, at least how many
hours does he need work on the weekends?
a
Ty works 22 hours in weekdays and 14 hours in weekends to earn at least $600 by working a total of 36 hours in a week.
In the question, we have to find how many hours he need work on the weekends.
Let he work x hours on weekdays.
Let he work y hours on weekends.
He works total 36 hours. So the required question is:
x + y = 36..................(1)
Ty earns $14 an hour working on weekdays. If he works x hours, then he earns total 14x.
Ty earns $21 an hour working on weekends. If he works y hours, then he earns total 21x.
He earns total $600.
So the required equation is:
14x + 21y = 600..................(2)
Multiply equation 1 by 21 and subtract by equation 2
21x + 21y - (14x + 21y) = 756 - 600
Simplify
7x = 156
Divide by 7 on both side, we get
x = 22.29
x = 22
Now put the value of x in equation 1, we get
22 + y = 36
Subtract 22 on both side, we get
y = 14
Hence, he works 22 hours in weekdays and 14 hours in weekends.
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A wheel 2. 0 m in diameter lies in the vertical plane and rotates about its central axis with a constant angular acceleration of 4 rad^-2 The wheel starts from rest at t=0 and the radius vector of a point A on the wheel makes an angle of 60º with the horizontal at this instant. Calculate the angular speed of the wheel, the angular position of the point A and the total acceleration at t=2s
The angular speed is 8 rad/s and the angular position of point A will be 8m/s².
The angular speed of the wheel at t=2s can be calculated using the equation ω=αt, where ω is the angular speed, α is the angular acceleration and t is the time. Substituting the given values, we get ω = 4 (2) = 8 rad/s.
The angular position of point A at t=2s can be calculated using the equation
θ = ωt + (1/2)αt².
Substituting the given values, we get
θ = 8 (2) + (1/2) (4) (2²) = 24 rad.
The total acceleration of point A at t=2s can be calculated using the equation a = rα, where a is the total acceleration, r is the radius of the wheel and α is the angular acceleration. Substituting the given values, we get a = 2 (4) = 8 m/s²
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a certain city divides naturally into ten district neighborhoods. how might a real estate appraiser select a sample of singlefamily homes that could be used as a basis for developing an equation to predict appraised value from characteristics such as age, size, number of bathrooms, distance to the nearest school, and so on? is the study enumerative or analytic?
The problem deals with the concept of simple random sampling and stratified random sampling techniques. Here, use these techniques to identify the correct statement for the given study.
What is a simple random sample?a sampling technique that guarantees each population member or object has the same chance of being selected for the sample.
Stratified random sample: A stratified random sample is created by first separating the population's components into distinct, non-overlapping groups, or strata, and then randomly choosing a sample from each stratum in turn. A stratified SRS is a unique type of stratified sampling that chooses units from each stratum using SRS.
To forecast appraised value from different variables, such as size, age, number of bathrooms, etc., the real estate appraiser may utilize basic random or stratified random samples.
A real estate appraiser must get a random sample of single-family homes in the basic random sample approach in order to analyze or anticipate the necessary qualities.
He must create a basic random sample from each of the ten neighborhood districts that are specified as stratums in the stratified random sample method. He can research or extrapolate the necessary qualities from the sample that was generated. The population in this investigation is limited and recognizable. This makes it enumerative.
An appropriate approach is to create a stratified random sample by selecting a simple random sample from each of the 10 district neighborhoods or to construct a random sample of all single-family residences in the city.
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Penelope has a smart phone data plan that costs $35 per month that includes 9 GB of data, but will charge an extra $25 per GB over the included amount. How much would Penelope have to pay in a month where she used 5 GB over the limit? How much would Penelope have to pay in a month where she used went over by xx GB?
For the 5 GB and x GB over use of the data plan per month the total cost paid by Penelope is $160 and $(35 + 25x ).
Let 'x' represents the extra GB used by Penelope in his smart phone data plan.
Cost to used fixed 9 GB of data in a month is equal to $35
Extra charge per GB over the included amount for fixed 9 GB is $25
Equation represents the data plan relation is:
35 + 25x
When Penelope used 5 GB extra over the 9 GB data plan then ,
x = 5GB
Amount to be paid for extra 5 GB is
= 35 + 25( 5 )
= 35 + 125
= $ 160
Amount to be paid for extra 'x' GB is $(35 + 25x).
Therefore, the total cost for extra use of data plan per month is:
a. For 5 GB it is $160.
b. For x GB it is $(35 + 25x).
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Which of the following mathematical properties is described by the statement, “whatever is done on one side of the equation must also be done on the other side of the equation”?
Identity Property of Addition
Associative Property
Communitive Property
Property of Equality
The property descibed by the statement "whatever is done on one side of the equation must also be done on the other side of the equation” is (d) Property of Equality
How to determine the property of the statement?From the question, we have the following parameters that can be used in our computation:
“whatever is done on one side of the equation must also be done on the other side of the equation”
This means that if x is added to the first side, it must be done on the other side
The property described above is (d)
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Select the correct answer. Which value of x makes this equation true?-12x-2(x+9)=5(x+4)
Answer: X= - 2
Step-by-step explanation:
Find the value of x rounded to the nearest tenth.
Answer: 21.3
Step-by-step explanation: Let us split the figure into two triangles. If the total bottom length is 22, than the triangle's base is 22/2=11
We can use the Pythagorean theorem to get: 11^2+x^2=24^2 or 121+x^2=576
576-121=455
sqrt 455=21.33................
rounds to 21.3
Can someone give me a step-by-step on how to solve this, please?
The rate of change over 1 ≤ x ≤ 3 is -5.76 and rate of change over 4 ≤ x ≤ 7 is -2.665.
What is a function?A relation is a function if it has only One y-value for each x-value.
The general form of an exponential function is:
f(x) = 40(0.8)ˣ
where 40 is the initial value and 0.8 is the decay factor.
To get the average rate of change over an interval, compute the slope of the line that connects the endpoints of the interval (it's called the secant line). For an interval running from x₁ to x₂, the slope formula for a straight line is:
slope = (f(x₂)-f(x₁)) / (x₂-x₁)
where f(x₁) and f(x₂) are the values of your function at x₁ and x₂.
So for the interval 4 ≤ x ≤ 7
slope = average rate of change = (f(7)-f(4)) / (7-4)
=8.388-16.384/3
=-2.665
For the interval 1 ≤ x ≤ 3
Average rate of change =f(3)-f(1)/2
=20.48-32/2
=-5.76
Hence, the Rate of change over 1 ≤ x ≤ 3 is -5.76 and rate of change over 4 ≤ x ≤ 7 is -2.665.
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The points B(-9,1), C(-4,-1), D(-2,4)B(−9,1),C(−4,−1),D(−2,4), and E(-7,7)E(−7,7) form quadrilateral BCDE. Plot the points then click the "Graph Quadrilateral" button.
All the correct expressions are,
Slope of BC = - 2/5
Slope of CD = 5/2
Slope of DE = - 3/5
Slope of EB = 3
Length of BC = √ 29
Length of CD = √ 29
Length of DE = √34
Length of EB = √40
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 have to given that;
Coordinate of B = (- 9, 1)
Coordinate of C = (- 4, - 1)
Coordinate of D = (- 2, 4)
Coordinate of E = (- 7, 7)
Now, We know that;
The slope of line passing through the points (x₁ , y₁) and (x₂, y₂) is,
⇒ m = (y₂ - y₁) / (x₂ - x₁)
Thus, Slope of BC is,
⇒ m = (- 1 - 1) / (- 4 - (-9))
⇒ m = - 2/5
Slope of CD is,
⇒ m = (4 - (-1)) / (- 2 - (-4))
⇒ m = 5/2
Slope of DE is,
⇒ m = (7 - 4) / (- 7 - (-2))
⇒ m = 3/- 5
⇒ m = - 3/5
Slope of EB is,
⇒ m = (1 - 7) / (- 9 - (-7))
⇒ m = - 6/-2
⇒ m = 3
Length of CD is,
⇒ CD = √ (- 2 - (-4))² + (4 - (- 1))²
= √4 + 25
= √ 29
Length of BC is,
⇒ BC = √ (- 4 - (-9))² + (- 1 - 1)²
= √25 + 4
= √ 29
Length of DE is,
⇒ DE = √ (- 2 - (-7))² + (4 - 7)²
= √25 + 9
= √ 34
Length of EB is,
⇒ EB = √ (- 7 - (-9))² + (7 - 1)²
= √4 + 36
= √ 40
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The table represents a proportional relationship. Write an equation to represent the relationship.
An equation to represent the relationship is y=15x.
What is linear equation ?
Linear equation can be defined as the equation in which highest degree is one.
Given ,
The table represents a proportional relationship.
So we have to write an equation to represent the relationship
Given,
Time that is
x = {5,8,12 }
Distance that is
y = {75,120,180}
so here from the given table,
75 = 15*5
120 = 8*15
180 = 12*15
hence , it is in the form of y = 15x.
Therefore, An equation to represent the relationship is y=15x.
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4x+3y=12 in y=mx+b form
Answer:
y = -4/3x + 4
Step-by-step explanation:
4x+3y=12
3y = -4x + 12
y = -4/3x + 4
Hello can someone kind help me with all my problems
There are 22 purple notebooks in the bin.
There are 37 pink socks in the bin.
How many copies have been sold to date?In order to determine the number of purple notebooks that are in the bin, we would develop an expression to multiply the total number of notebooks (T) by the percentage of the number of purple notebooks placed in the note bin as follows;
Total number of purple notebooks (P) = 25/100 × 88
Total number of purple notebooks (P) = 0.25 × 88
Total number of purple notebooks (P) = 22 notebooks.
Question 2.Total number of socks (S) = 50/100 × 74
Total number of socks (s) = 0.50 × 74
Total number of socks (s) = 37 socks.
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What is the domain of the function f(x) = -x +15?
O x ≤ 30
Ο x > -2
O all real numbers
O x >0
Answer: The domain of a function is the set of all input values (x-values) for which the function produces a valid output. In the case of the function f(x) = -x + 15, the function will take any real number as input and produce a corresponding output.
Therefore, the domain of the function f(x) = -x + 15 is all real numbers. This can be represented as (-∞, ∞) or simply as "all x".
Option O is correct: x are all real numbers
Option O x ≤ 30, O x > -2, O x >0 are incorrect
Step-by-step explanation:
Please help with this problem!
True. Testing other points in addition to the vertices is necessary
During optimization of a function, must you test other points in addition to the vertices?True. When trying to optimize (find the max or min) of a function, it is not enough to test only the vertices. The function may have a maximum or minimum value at a point that is not a vertex, such as an inflection point, or a point on the edge of the domain.
Thus, testing other points in addition to the vertices is necessary in order to ensure that the maximum or minimum value of the function has been found.
Linear programming is a powerful tool that can be used to model and solve a wide range of real-world problems in fields such as finance, manufacturing, transportation, and resource allocation.
For example, it can help to determine the optimal number of products to produce, the best way to use resources (e.g. labor, materials) and the best scheduling to minimize costs and maximize profits.
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if a fair penny is tossed three times and comes up heads all three times, the probability of heads on the fourth trial is . a. smaller than the probability of tails b. larger than the probability of tails c. .0625 d. .50
The probability of obtaining heads on fourth trial is 1/2
In a random experiment of tossing a penny, let H be the event of obtaining head and T be the event of obtaining tail
As it is a fair penny, the probability of obtaining head=1/2
The outcomes of the trials are independent when a coin is tossed multiple times.
Probability of getting three heads ,
P(3H)=P(H∩H∩H)
=P(H)xP(H)xP(H)
=1/2x1/2x1/2
=1/8
We will use conditional probability to find the probability of obtaining head in the fourth trial, given that three heads are already obtained in three trials
P(H|3H)=P(H∩3H)/P(3H)=(1/8x1/2)/(1/8)=1/2
Thus, the required probability is 1/2
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If you have a different linear equation such as y = 1 2 x + 4 , the slope and y intercept would be which of the following? Question 5 options: slope: 2 y-intercept: 4 slope: 4 y-intercept: 1 2 slope: - 1 2 y-intercept: 4 slope: 1 2 y-intercept: 4
The slope of the line is 1/2 and the y-intercept is 4. Then the correct option is D.
What is a linear equation?A connection between a number of variables results in a linear model when a graph is displayed. The variable will have a degree of one.
The linear equation is given as,
y = mx + c
Where m is the slope of the line and c is the y-intercept of the line.
If you have a different linear equation such as y = (1/2) x + 4. Then the slope and y-intercept of the line are given as,
m = 1/2
y = 4
The slope of the line is 1/2 and the y-intercept is 4. Then the correct option is D.
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3 to the power of a+4
The word description "3 to the power of a + 4" should be written as 3^{a + 4}.
What is an exponent?In Mathematics, an exponent is sometimes referred to as power and it can be defined as a mathematical operation that is used in conjunction with an algebraic expression to raise a quantity to the power of another and it is generally written as;
bⁿ
Where:
the variables b and n represent numerical values or an algebraic expression.
By applying the property of exponents based on the law of indices for powers, we have the following:
Expression = 3^{a + 4}.
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Complete Question:
What is 3 to the power of a+4?
d.
What do you notice about the graph of the quadratic equation and where it crosses the
x-axis? How do these values compare to the solutions of x2 - 6x + 5 = 0? Explain.
This is consistent with the solutions for the equation x2 - 6x + 5 = 0 being x = 1 and x = 5.
What is equation?An equation is a mathematical statement that expresses the equality of two expressions. It consists of one or more terms, each of which contains one or more variables. The terms are separated by an equal sign (=) and are usually written in a standard form known as an algebraic equation. Equations can be used to model and solve problems in a wide variety of fields, from physics and chemistry to economics and finance.
The graph of the quadratic equation is a parabola that opens upward and intersects the x-axis at two points. The x-intercepts occur when the equation is equal to zero, so the x-intercepts for the equation x2 - 6x + 5 = 0 are the solutions for x. These solutions are x = 1 and x = 5.
The graph of the quadratic equation also shows that the two solutions of the equation, x = 1 and x = 5, are the x-intercepts of the parabola. This means that when x equals 1 or 5, the equation is equal to zero, so the x-intercepts are the solutions of the equation. This is consistent with the solutions for the equation x2 - 6x + 5 = 0 being x = 1 and x = 5.
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Solve the equation. 1.2=1.5k I NEED HELP BADDDD
Answer: 0.3K IS THE DIFFERENCE
Step-by-step explanation:
HOPE IT HELPS =D
Answer:
3k
Step-by-step explanation:
1. Combine multiplied terms into a single fraction
2. Multiply all terms by the same value to eliminate fraction denominators
3. Cancel multiplied terms that are in the denominator
4. Multiply the numbers
Suppose the demand for a product can
expressed as p(x)=0.2x² +1.13x+5.2
where x is given in units of a thousand.
A. Find the average rate of change of
demand when the number of items
demanded increases from 2 thousand
units to 4 thousand units.
B. Find the average rate of change of
demand when the number of items
demanded increases from 1 thousand
units to 5 thousand units.
The average rate of change of demand when the number of items demanded increases from 2 thousand units to 4 thousand units is 401.13 and if it increase from 1 thousand units to 5 thousand units is 201.13
What is Algebraic expression ?
Algebraic expression can be defined as the combination of variables and constants.
Given,
The expression, p(x) = 0.2 x^2 + 1.13x+5.2
where x is given in units of a thousand .
The average rate of change of demand when the number of items demanded increases from 2 thousand units to 4 thousand units.
p(x) = 0.2x^2 + 1.13x+5.2
p(2000 ) = 0.2 (2000*2000) + 1.13 (2000) + 5.2
= 800000 + 2260 + 5.2
= 802265.2
p(4000 ) = 0.2 (4000*4000 ) + 1.13 (4000) + 5.2
= 1600000 + 4520 + 5.2
= 1604525.2
p(2000) - p (4000) / 2000-4000 = 802265.2 - 1604525.2 / -2000
= -8,02,260 / -2000
= 401.13
p (1000) = 0.2 (1000*1000) + 1.13 (1000) + 5.2
= 200000 + 1130 + 5.2
= 201135.2
p (5000 ) = 0.2 (5000*5000) + 1.13(5000) + 5.2
= 1000000 + 5650 + 5.2
= 1005655.2
p (1000) - p (5000) / 1000 - 5000 = 201135.2 - 1005655.2 / -4000
= 804520 / 4000
= 201.13
Hence, The average rate of change of demand when the number of items demanded increases from 2 thousand units to 4 thousand units is 401.13 and if it increase from 1 thousand units to 5 thousand units is 201.13.
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The sum of money of $3 500 is divided among Adrian, Sean and James. Sean received half. Adrian received $850 and James received the remainder. Calculate:
(a) Sean's share
(b) James' share
c) the ratio in which the $3500 was divided among the three person
Answer:
a. 1750; b. 900; c. sean 50%, Adrian = 850/3500= 24.3%, James =25.7%
Step-by-step explanation:
Sean got half = 3500/2 =1750
Adrian = 850
James = 3500-1750-850 =900
If 6x/x^2-3x is subtracted from the sum of 3/5x-15 and x+3/x-3 what is the result?
If 6x/(x²-3x) is subtracted from the sum of 3/(5x-15) and (x+3)/(x-3) , then the result is (12 - 5x)/(x - 3) .
The sum of the algebraic expressions 3/(5x-15) and (x+3)/(x-3) are written as ; ⇒ 3/(5x-15) + (x+3)/(x-3) ;
taking 5 common from the denominator of the first term ,
we get ;
⇒ 3/5(x-3) + (x+3)/(x-3) ;
taking LCM as 5(x-3) and simplifying further ,
we have ;
⇒ 3/5(x-3) + (x+3)/(x-3) ;
⇒ [3 + 5(x+3)]/5(x-3) ;
⇒ (5x + 18)/5(x-3) .......equation(1)
Now 6x/(x²-3x) is subtracted from the equation(1)
⇒ (5x + 18)/5(x-3) - 6x/(x²-3x) ;
⇒ (5x + 18)/5(x-3) - 6x/x(x-3) ;
⇒ (5x + 18)/5(x-3) - 6/(x-3) ;
taking LCM as as 5(x-3) and simplifying further ,
we have ;
⇒ (5x + 18 - 30)/5(x-3) ;
⇒ (5x -12)/5(x-3) ;
Therefore , the final result is (5x -12)/5(x-3) .
The given question is incomplete , the complete question is
If 6x/(x²-3x) is subtracted from the sum of 3/(5x-15) and (x+3)/(x-3) . What is the result ?
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How do I figure out 1.5+5x
Answer:
5
Step-by-step explanation:
1.5+5x
The derivative of a polynomial is the sum of the derivatives of its terms. The derivative of a constant term is 0. The derivative of ax n is nax n−1
5x1−1
Subtract 1 from 1.
5x 0
For any term t except 0, t 0 =1.
5×1
For any term t, t×1=t and 1t=t.
5
This is how i would solve it :))
Need help for this question
Answer:
Step-by-step explanation: Hello! You know that there are 3 students who scored 55, 4 students who scored 60, 8 students who scored 65, 4 students who scored 70, 4 students who scored 75, 1 student who scored 80, 9 students who scored 85, and 5 students who scored 90. To find the mean of a set of values, here is the formula:
a + b + c + d + e . . .
--------------------------
2
Each variable should represent the score of a single student, so write each score for each individual student.
(55 + 55 + 55 + 60 + 60 + 60 + 60 + 65 + 65 + 65 + 65 + 65 + 65 + 65 + 65 + 70 + 70 + 70 + 70 + 75 + 75 + 75 + 75 + 80 + 85 + 85 + 85 + 85 + 85 + 85 + 85 + 85 + 85 + 90 + 90 + 90 + 90 + 90)/2
You could divide all of this by 2 or you could multiply the values and then add them:
((55 x 3) + (60 x 4) + (65 x 8) + (70 x 4) + (75 x 4) + 80 + (85 x 9) + (90 x 4))/2
Either way works, but the second way is probably faster.
--------------------------------------------------------------------------------------------------
How to find the answer:
55 x 3 = 165
60 x 4 = 240
65 x 8 = 520
70 x 4 = 280
80 = 80
85 x 9 = 765
90 x 4 = 360
Substitute this into the equation
(165 + 240 + 520 + 280 + 80 + 765 + 360)/2
(405 + 800 + 845 + 360)/2
(1205 + 1205)/2
2410/2
= 1205
The answer is 1205
please help
what are the areas of these figures ?
the area of the triangle is____cm^2
the area of the rectangle is___cm^2
The areas of the given triangle and rectangle are 60 cm² and 120 cm² respectively.
What are a triangle and rectangle?
A triangle is a three-sided polygon which has 3 angles formed when two sides join to form a vertex.
A rectangle is a four-sided polygon whose opposite sides have the same lengths and all sides are perpendicular to each other.
The given triangle is a right-angled triangle.
The area of a right-angled triangle is 1/2*b*h
where,
b= base of the triangle
h = height of the triangle
From the figure, b = 10 cm and h = 12 cm
Area of triangle = 1/2*10*12 = 60 cm²
The given rectangles have a length of 12 cm and a breadth of 10cm.
The area of the rectangle is given by the product of breadth and length.
Area of rectangle = 12 * 10 = 120 cm²
Therefore the areas of the given triangle and rectangle are 60 cm² and 120 cm² respectively.
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Can you help me I really need help
In the given situation the given values can be correctly modeled by a (B) linear function.
What is a linear function?
A linear function has a straight line as its graph. A linear function has the form shown below. a + bx = y = f (x).
A linear function consists of one independent variable and one dependent variable.
The independent and dependent variables are X and Y, respectively.
In mathematics, the terms "linear function" refers to two distinct but connected concepts: In calculus and related subjects, a polynomial function of degree zero or one with a graph that is a straight line is referred to as a linear function.
Therefore, in the given situation the given values can be correctly modeled by a (B) linear function.
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B51 Find the number of ten-letter combinations (allowing repetition) from the set A, B,C that contain: (a) at least one of each letter; (b) at least two of each letter; (c) at least one A, two Bs, and three Cs; (d) any number of each letter; (e) at least one A and at least two Bs
As per the combination method,
(a) at least one of each letter is 36
(b) at least two of each letter is 15
(c) at least one A, two Bs, and three Cs is 15
(d) any number of each letter is 120
(e) at least one A and at least two Bs is 36
In math the term combination method is known as the number of possible combinations that can be obtained by taking a sample of items from a larger set.
Here we have given that the value of n is 10 and the value of r is 3.
Then the resulting values are calculated as,
Here for the letters with at least one of each letter then the length is 10 with a 3 elements to choose from the given set.
Here we know that 3 of the 10 slots will be taken up by A,B, and C. With 7 remaining, any letter could take up those spots it will be calculated as,
=> (7+3-1 choose 3-1)
=> (9 choose 2).
=> 36
Where as the other one is with at least two of each letter
Then here we have the letter with two of each letter 6 letters will take up 10 slots no matter what is calculated as,
=> 10-6 = 4
Now then we have 3 to choose from so (4 + 3 - 1 choose 3-1).
=> (6 choose 2)
=> 15
And the another option is to have at least one A, two Bs, and three Cs
=> 1A + 2B + 3C will take up 6 spots.
=> (4+ 3-1 choose 3-1)
=> (6 choose 2).
=> 15
Where as the count is any number of each letter
=> 10 choose 3.
=> 120
Finally if we take at least one A and at least two Bs
Then at least 1 A and two Bs taking up a total of 3 slots is calculated as,
=> 10 - 3 = 7.
=> (7 + 3 - 1 choose 3 - 1)
=> (9 choose 2) = 36
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