The mean of the temperatures is 29°C, which corresponds to answer B.
Step-by-step explanation:1. Add all the values together.For convenience reasons we are going to avoid the use of the Celsius symbol (°C) during the calculations.
[tex]19+22+30+45=116[/tex]
2. Count the amount of given values.There are 4 values of temperature.
3. Divide the total of the sum by the amount of given values.[tex]\frac{116}{4}=29[/tex]
4. Conclude.The mean of the temperatures is 29°C, which corresponds to answer B.
if you form a triangle from three given side lengths, will you always get one triangle or more than one triangle?
If the three side lengths are all different then you will only get one triangle. However, if two or more of the side lengths are the same then you can get more than one triangle.
What is known as a triangle?A triangle is a polygon with three vertices and three sides. The angles of the triangle are formed by the connection of the three sides end to end at a point. The triangle's three angles add up to 180 degrees in total
How is a triangle made?Three diagonals must meet at three different locations to form a triangle. There are five possible arrangements of three diagonals. We categorize them according on how many different diagonal endpoints there are. The quantity of triangles with 3, 4, and 5 endpoints will be counted directly.
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Need an answer pls will mark Brainliest
Answer:
4 m^3 - 38 m^2 + 57 m - 72 = (4 m^2 - 6 m + 9)(m - 8) + 0
Step-by-step explanation:
Compute (4 m^3 - 38 m^2 + 57 m - 72) ÷ (m - 8) using polynomial long division:
m - 8 | 4 m^3 | - | 38 m^2 | + | 57 m | - | 72
To eliminate the leading term of the numerator, 4 m^3, multiply m - 8 by 4 m^2 to get 4 m^3 - 32 m^2. Write 4 m^2 on top of the division bracket and subtract 4 m^3 - 32 m^2 from 4 m^3 - 38 m^2 + 57 m - 72 to get -6 m^2 + 57 m - 72:
| | | 4 m^2 | | | |
m - 8 | 4 m^3 | - | 38 m^2 | + | 57 m | - | 72
| -(4 m^3 | - | 32 m^2) | | | |
| | | -6 m^2 | + | 57 m | - | 72
To eliminate the leading term of the remainder of the previous step, -6 m^2, multiply m - 8 by -6 m to get -6 m^2 + 48 m. Write -6 m on top of the division bracket and subtract -6 m^2 + 48 m from -6 m^2 + 57 m - 72 to get 9 m - 72:
| | | 4 m^2 | - | 6 m | |
m - 8 | 4 m^3 | - | 38 m^2 | + | 57 m | - | 72
| -(4 m^3 | - | 32 m^2) | | | |
| | | -6 m^2 | + | 57 m | - | 72
| | | -(-6 m^2 | + | 48 m) | |
| | | | | 9 m | - | 72
To eliminate the leading term of the remainder of the previous step, 9 m, multiply m - 8 by 9 to get 9 m - 72. Write 9 on top of the division bracket and subtract 9 m - 72 from 9 m - 72 to get 0:
| | | 4 m^2 | - | 6 m | + | 9
m - 8 | 4 m^3 | - | 38 m^2 | + | 57 m | - | 72
| -(4 m^3 | - | 32 m^2) | | | |
| | | -6 m^2 | + | 57 m | - | 72
| | | -(-6 m^2 | + | 48 m) | |
| | | | | 9 m | - | 72
| | | | | -(9 m | - | 72)
| | | | | | | 0
The quotient of (4 m^3 - 38 m^2 + 57 m - 72)/(m - 8) is the sum of the terms on top of the division bracket. Since the final subtraction step resulted in zero, m - 8 exactly divides 4 m^3 - 38 m^2 + 57 m - 72, and there is no remainder.
| | | 4 m^2 | - | 6 m | + | 9 | (quotient)
m - 8 | 4 m^3 | - | 38 m^2 | + | 57 m | - | 72 |
| -(4 m^3 | - | 32 m^2) | | | | |
| | | -6 m^2 | + | 57 m | - | 72 |
| | | -(-6 m^2 | + | 48 m) | | |
| | | | | 9 m | - | 72 |
| | | | | -(9 m | - | 72) |
| | | | | | | 0 | (remainder) invisible comma
(4 m^3 - 38 m^2 + 57 m - 72)/(m - 8) = (4 m^2 - 6 m + 9) + 0
Write the result in quotient and remainder form:
Answer: 4 m^3 - 38 m^2 + 57 m - 72 = (4 m^2 - 6 m + 9)(m - 8) + 0
Determine whether the expressions are equivalent.
0.4(k+3) is to 0.4 k+1.2
Equivalent. 0.4(k+3)=0.4 k+1.2
Not equivalent. 0.4(k+3)=0.4 k+0.8
Equivalent. 0.4(k+3)=0.4 k+0.4
Not equivalent. 0.4(k+3)=0.4 k+0.4
Both the expressions are equivalent to each other.
What do you mean by algebraic expression?
The concept of algebraic expressions is the use of letters or alphabets to represent numbers without providing their precise values. We learned how to express an unknown value using letters like x, y, and z in the fundamentals of algebra. Here, we refer to these letters as variables. Variables and constants can both be used in an algebraic expression. A coefficient is any value that is added before a variable and then multiplied by it.
Given expression:
LHS
0.4(k+3)
= 0.4k + 0.4 × 3
= 0.4k + 1.2
RHS
0.4 k+1.2
LHS = RHS
Therefore, both the expressions are equivalent to each other.
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Suppose that the functions g and h are defined for all real numbers x as follows.
g(x)=x-6
h(x)=4 x+3
By using the concept of function, it can be calculated that-
(h.g)(x) = 4x - 21
(h + g)(x) = 5x - 3
(h - g)(-1) = 6
What is a function?
A function from A to B is a rule that assigns to each element of A a unique element of B. A is called the domain of the function and B is called the codomain of the function.
There are different operations on functions like addition, subtraction, multiplication, division and composition of functions.
Here,
a) The given functions are
g(x)= x - 6
h(x) = 4x + 3
(h.g)(x) = h(g(x)) = 4(x - 6) + 3 = 4x - 21
(h + g)(x) = h(x) + g(x) = 4x + 3 + x - 6 = 5x - 3
(h - g)(x) = h(x) - g(x) = 4x + 3 - x + 6 = 3x + 9
(h - g)(-1) = 3 [tex]\times[/tex] (-1) + 9 = 6
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f(x) = 2(x²-7)
Find f(2).
everything at a store is on sale. The store offers a 25% discount. The regular price of a t-shirt is 20%. What is the discount price? Please help me.
The discount price of the shirt will be $15.
How to calculate the price?A percentage is a value or ratio that may be stated as a fraction of 100. If we need to calculate a percentage of a number, we should divide it's entirety and then multiply it by 100. The percentage therefore refers to a component per hundred. Per 100 is what the word percent means. It is represented by %.
From the information, the store offers a 25% discount and the regular price of a t-shirt is $20. Therefore, based on the information illustrated, the main points to note will be:
Discount = 25%
Regular price = $20.
In this case, the discount price will be the difference between the regular price and the discount.
The discount price will be:
= Regular price - Discount
= $20 - ($20 × 25%)
= $20 - $5
= $15
The price is $15.
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Complete question
everything at a store is on sale. The store offers a 25% discount. The regular price of a t-shirt is $20. What is the discount price? Please help me
Fill in the diagram based on the following information
30% of the total is 12 grams
in this given question 30% of 12 will be 3.6.
What is the percentage?
A percentage that represents a tenth of a quantity. One percent, denoted by the symbol 1%, is equal to one-hundredth of something; hence, 100 percent denotes the full thing, and 200 percent designates twice the amount specified. A portion per hundred is what the percentage denotes. The percentage refers to one in a hundred. The % sign is used to denote it.
30% of the total is 12 grams
So the given 30% of 12 will be
12*30/100
=3.6
Hence, in this given question 30% of 12 will be 3.6.
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Solve the equation for w.
w divided by 1.1 minus 3.9 equals 10.5
6
7.26
13
15.84
W will be 15.84.
What is an equation?The definition of an equation in algebra is a mathematical statement proving the equality of two mathematical expressions. For instance, the equation 3x + 5 = 14 has the two expressions 3x + 5 and 14 that are separated by the 'equal' sign.
We know that w divided by 1.1 minus 3.9 equals 10.5
That will be W/1.1 - 3.9 = 10.5
W/1.1 = 10.5 + 3.9
W/1.1 = 14.4
W = 14.4* 1.1
W = 15.84
W will thus equal 15.84.
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B OA) A Which statements are necessary to prove the two triangles are congruent by SSS? O B) D =CD, AB = C, mzBDA = 90° AD, BD = CD, mzBDA = 90° O a OD) AD AD, BD = CD, AB = AC BD = CD, AB= AC
The statements that are necessary to prove the two triangles are congruent by SSS are AD≅ AD, BD ≅ CD, and AB ≅ AC.
What is congruence?
In geometry, two figures or objects are said to be congruent if their shapes and sizes match, or if one is the mirror image of the other.
Here,
To prove that two triangles are congruent using the SSS Congruence criterion, we must establish that all three sides of one triangle are equal to all the corresponding sides of the other.
Thus,
Side AD in ∆ADB is congruent to side AD in ∆ADC.
Side BD in ∆ADB is congruent to side CD in ∆ADC.
Side AB in ∆ADB is congruent to side AC in ∆ADC.
Hence, based on the SSS congruence criterion, it is concluded that both triangles are congruent. i.e AD≅ AD, BD ≅ CD, AB ≅ AC.
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Answer:
D
Step-by-step explanation:
To prove that two triangles are congruent using the SSS Congruence criterion, we must establish that all three sides of one triangle are equal to all the corresponding sides of the other.Thus,Side AD in ∆ADB is congruent to side AD in ∆ADC.Side BD in ∆ADB is congruent to side CD in ∆ADC.Side AB in ∆ADB is congruent to side AC in ∆ADC.Therefore, based on the SSS congruence criterion, it is concluded that both triangles are congruent.The statements necessary to prove that they are congruent by SSS is: D
Use the formula A=P(1+r/n)ⁿt to solve the compound interest problem
Find how long it takes for $ 900 to double if it is invested at 6 % interest compounded monthly.
The money will double in value in approximately years.
Answer:
11.6 years
Step-by-step explanation:
1800 = 900 x (1 + 0.06/12)^ 12t
1800 = 900 x (1 + 0.005)^12t
1800 = 900 x (1.005)^12t
1800 = 900 x 1.061678^t
1800/900 = (900 x 1.061678^t)/ 900
2 = 1.061678^t
log(1.061678) 2 = t
t = 11.5813
t = 11.6
A new computer game costs $160,000 to research and develop. Once completed, individual games can be produced for just $.20 each. If the first 275 are given away as samples, how many games must be sold for the average price to be $2?
Answer: The correct answer is (d) 89,200 games.
To find the number of games that must be sold, we need to set up an equation to represent the total revenue from the game sales and solve for the number of games sold.
The total revenue from the game sales is equal to the number of games sold multiplied by the price per game, minus the cost of research and development.
Let X be the number of games sold
Total revenue = X * $2 - $160,000
We also know that the total cost of producing the games is equal to the number of games produced multiplied by the cost per game.
Total cost = X * $0.20
We can set the total revenue equal to the total cost and solve for X:
X * $2 - $160,000 = X * $0.20
X = 89,195
Since we gave away the first 275 games for free, we need to add 275 to the number of games sold to find the total number of games that must be sold.
Total number of games sold = X + 275 = 89,195 + 275 = 89,470
Therefore, the correct answer is (d) 89,200 games.
NEED HELP NOW Convert 110 kilograms to pounds. (1 kilogram = 2.204 pounds)
The required answer in the given problem, given weight of 5 kilograms and convert it to 11.02 pounds. By multiplying the given weight by the conversion factor of 2.204 pounds/kilogram, obtain the weight in pounds.
Conversion from kilograms to pounds:
To convert a weight value from kilograms to pounds, we use the conversion factor of 1 kilogram equals 2.204 pounds. This means that for every 1 kilogram, there are 2.204 pounds. To convert a weight from kilograms to pounds, we multiply the given value in kilograms by the conversion factor.
To convert 110 kilograms to pounds, we can use the conversion factor that states 1 kilogram is equal to 2.204 pounds. Here's the step-by-step process:
Step 1: Write down the given weight in kilograms, which is 110 kilograms.
Step 2: Multiply the given weight by the conversion factor:
110 kilograms x 2.204 pounds/kilogram
Step 3: Perform the multiplication:
110 x 2.204 = 242.44 pounds
Therefore, 110 kilograms is equal to 242.44 pounds.
In the given problem, given weight of 5 kilograms and convert it to 11.02 pounds. By multiplying the given weight by the conversion factor of 2.204 pounds/kilogram, obtain the weight in pounds.
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A wire is first bent into the shape of a rectangle with width 5cm and length 16 cm . Then the wire is unbent and reshaped into a triangle. If each side of the triangle has equal length, what is this length?
Hello,
I hope you and your family are doing well!
To find the length of each side of the triangle, we can use the formula for the perimeter of a rectangle: perimeter = 2 * (width + length). In this case, the perimeter of the rectangle is 2 * (5cm + 16cm) = 42 cm. Since each side of the triangle has the same length, and the perimeter of the triangle is equal to the perimeter of the rectangle, the length of each side of the triangle is 42cm / 3 = 14 cm.
Therefore, the length of each side of the triangle is 14 cm.
---
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What is the solution to 9|x – 8| < 36?
4 < x < 12
x < –4 or x > 12
x > –12 or x < 8
–4 < x < 8
Answer:
4 < x < 12
Step-by-step explanation:
x - 8 ≥ 0
x ≥ 8
If x ≥ 8
9(x-8) < 36
9x - 72 < 36
9x < 36 + 72
9x < 108
9/9 x < 108/9
x < 12
8 ≤ x < 12
if x < 8
9(-x+8) < 36
-9x + 72 < 36
-9x < 36 - 72
-9x < -36
-9/-9 x > -36/-9
x > 4
4 < x < 8
4<x<12
Multiply by using the Distributive Property. (x2 + 3)(x − 4) =
The expression is obtained by simplifying it by the distributive Property is x³-4x²+3x-12.
What is distributive property?The distributive property is the property that expresses the distributive law of algebraic multiplication. Similarly, in polynomial expression we can use the distributive property, for example, we can write a(c+b) in the form of (ac+bc).
It is given that, the expression is (x² + 3)(x − 4)
The expression is simplified using the distributive property as,
=(x² + 3)(x − 4)
=x²(x − 4)+3(x − 4)
=x³-4x²+3x-12
Thus, the expression is obtained by simplifying it by the distributive Property is x³-4x²+3x-12.
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The multiplication of the term (x² + 3)(x - 4) using distributive property is x³ - 4x² + 3x - 12.
Distributive property explains to us how to solve expressions in the form of a(b + c).
After apply distributive property, this term converts as ab+ac
The given expression is,
(x² + 3)(x - 4)
Solve by using distributive property formula (a+b)(c+d)= a(c+d) + b(c+d),
x²(x - 4) + 3(x - 4)
Now further simplify by using formula a(b+c) = ab+ac,
x³ - 4x² + 3x - 12.
The required multiplication of the given terms is x³ - 4x² + 3x - 12.
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Many investors and financial analysts believe the Dow Jones Industrial Average (DJIA) gives a good barometer of the overall stock market. On January 31, 2006, 9 of the 30 stocks making up the DJIA increased in price (The Wall Street Journal, February 1, 2006). On the basis of this fact, a financial analyst claims we can assume that 30% of the stocks traded on the New York Stock Exchange (NYSE) went up the same day.
A sample of 72 stocks traded on the NYSE that day showed that 29 went up.
You are conducting a study to see if the proportion of stocks that went up is is significantly more than 0.3. You use a significance level of a = 0.02.
What is the test statistic for this sample?
Use p rounded to 4 decimal places. (Report answer accurate to 4
decimal places.)
test statistic =
What is the p-value for this sample? (Report answer accurate to 4 decimal places.)
p-value =
The p-value is either less than (or equal to) or
greater than?
This p-value leads to a decision to...
reject the null or accept the null or fail to reject the null?
As such, the final conclusion is that... (select a letter)
A) There is sufficient evidence to warrant rejection of the claim that the proportion of stocks that went
up is is more than 0.3.
B) There is not sufficient evidence to warrant rejection of the claim that the proportion of stocks that
went up is is more than 0.3.
C) There is sufficient evidence to support the claim that the proportion of stocks that went up is is more
than 0.3.
D) There is not sufficient sample evidence to support the claim that the proportion of stocks that went up
is is more than 0.3.
The test statistic and the p-value of the research are respectively z = 1.9072 and p-value = 0.0283 and the conclusion is that; D) There is not sufficient sample evidence to support the claim that the proportion of stocks that went up is is more than 0.3.
How to find the p-value from the research?Let us first of all define the hypotheses;
Null Hypothesis; H₀: p ≤ 0.3
Alternative Hypothesis; Hₐ: p > 0.3
Sample proportion is;
p^ = 29/72 = 0.403
The formula for the test statistic is;
z = (p^ - p)/(√(p(1 - p)/n))
Plugging in the relevant values gives;
z = (0.403 - 0.3)/(√(0.3(1 - 0.3)/72))
z = 1.9072
From p-value from z-score calculator, we have that;
p-value = 0.0283
p-value is greater than the significance value and as such we fail to reject the null hypothesis and conclude that There is not sufficient sample evidence to support the claim that the proportion of stocks that went up
is is more than 0.3.
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A waterproof speaker sells for $179, which is marked up by 30% of the selling price. What is the cost of the speaker? A. $152.30 B. $132.50 C. $123.50 D. $125.30
The original cost of the speaker will be equal to $125.3. The correct option is D.
What is the percentage?The percentage is defined as representing any number with respect to 100. It is denoted by the sign %. The percentage stands for "out of 100." Imagine any measurement or object being divided into 100 equal bits.
Given that a waterproof speaker sells for $179, which is marked up by 30% of the selling price.
The cost price of the speaker will be calculated as,
Cost price = 179 x 0.7
Cost price = $125.3
Therefore, the original cost of the speaker will be equal to $125.3. The correct option is D.
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Does it make sense to discuss the similitude between 2 rombii, 2 squares, 2 rectangles, 2 parralelegrams, and 2 conguent polygons
Yes, it makes sense to discuss the similarity between these shapes.
How to determine if it make sense to discuss the similitude between 2 rombii, 2 squares?Two shapes are considered similar if they have the same shape, but possibly different sizes. This means that the angles of the shapes are the same and the sides of the shapes are in the same ratio.
Two rhombi are similar if they have the same angle measures and the lengths of their sides are in the same ratio.
Two squares are similar if they have the same angle measures and the lengths of their sides are in the same ratio.
Two rectangles are similar if they have the same angle measures and the lengths of their sides are in the same ratio.
Two parallelograms are similar if they have the same angle measures and the lengths of their sides are in the same ratio.
Two congruent polygons are similar by definition, since congruent shapes are identical in size and shape.
It is important to note that similarity is different from congruence. Congruent shapes are identical in size and shape, whereas similar shapes have the same shape but possibly different sizes.
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An observer whose eye level is 5 ft is observing the top of an object with a mirror on the
ground. The mirror is 3 feet from the observer and 12 feet from the object. Calculate the
height of the object.
The height of the object is 20 feet.
What is Proportion?In general, the term "proportion" refers to a part, share, or amount that is compared to a total.
According to the concept of proportion, two ratios are in proportion when they are equal.
A mathematical comparison of two numbers is called a proportion. According to proportion, two sets of provided numbers are said to be directly proportional to one another if they increase or decrease in the same ratio. "::" or "=" are symbols used to indicate proportions.
Given:
An observer whose eye level is 5 ft is observing the top of an object with a mirror on the ground.
The mirror is 3 feet from the observer and 12 feet from the object.
Using proportion
5/ 3= x/ 12
60= 3x
x = 20 feet
hence, the height of the object is 20 feet.
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The sum of 7x^2+x+1 and -6x^2-8x-10
Answer:
Step-by-step explanation:
To find the sum of 7x^2+x+1 and -6x^2-8x-10, you can add the two expressions by combining like terms.
7x^2+x+1 + (-6x^2-8x-10)
= (7x^2 + -6x^2) + (x + -8x) + (1 + -10)
= x^2 + -7x + -9
Therefore, the sum of 7x^2+x+1 and -6x^2-8x-10 is x^2 - 7x - 9.
Fully simplify. -15xy(15x^5y^3)
Answer:
−225x^6y^4
Step-by-step explanation:
The number of bacteria in a certain population increases Accor to a continuous growth model,with a growth parameter of 9.6% per hour. How many hours does it take for the size of the sample to double
The required number of hours does it take for the size of the sample to double is 11 hours.
What is percentage?Percentages are basically divisions where the denominator is 100. To show that a number is a percent, we utilize the percent image (%) close to the number. For instance, in the event that you got 75 inquiries right out of 100 on a test (75/100), you would have scored 75%.
According to question:Let initial population be x.
As question says number of bacteria increases by 9.6% per hours.
9.6% of x takes 1 hours,
So, in 11 hours size of sample become 105.6%, means almost double of initial size.
Thus, 11 hours will be taken to make size double.
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Mario makes 6 throw shots out of his first 10 attempts
Ok⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
Answer: 4
Step-by-step explanation: If you are trying to see how many Mario missed that would be 4
Solve for x Round to the nearest tenth, if necessary.
Answer:
1.7
Step-by-step explanation:
tan30°=1/XX=1/tan30°X=1/0.57741.73191.7(to the nearest tenth)Two machines produce sound pressure levels of approximately 80 dB(A) each. What is their approximate combined sound pressure level?
83.01dB is approximate combined sound pressure level.
What is sound pressure level?
The result of the pressure changes in the air caused by sound waves is sound pressure, often known as sound pressure level. The hearing threshold is the lowest sound pressure that a human can hear, and the pain threshold is the maximum sound pressure that a person can tolerate.
The difference between sound pressure at the pain threshold and the hearing threshold is a million times higher.
We have sound pressure level denoted as (Lp) of each machine
i.e 80 dB,(given).
Therefore, we can calulate sound pressure level from both machines combined by given formula,
SPL = 10 * Log(10^(SPL(1) /10) + 10^(SPL(2) /10) +10^(SPL(3) /10) + ............)
where,
SPL= sound pressure level(dB)
=> SPL = 10 * Log(10^(80 /10) + 10^(80 /10))
=> SPL = 10 * Log(10^8 + 10^8)
=> SPL = 10 * 8.301
=> SPL = 83.01dB
Hence, 83.01dB is approximate combined sound pressure level.
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Describe how the values of r, x, and y relate to each of sinø, cosø and tanø
the values of r, x, and y are related to the sine, cosine, and tangent of angle ø as:
sin ø = y / r
cos ø = x / r
tan ø = y / x
What is trigonometry?
Trigonometry is a branch of mathematics that studies relationships between side lengths and angles of triangles.
In a right triangle, the sides are typically labeled as follows:
The side opposite the angle ø is called the "opposite side" and is labeled "y".
The side adjacent to angle ø is called the "adjacent side" and is labeled "x".
The hypotenuse is the side opposite the right angle and is labeled "r".
The values of r, x, and y can be used to calculate the sine, cosine, and tangent of angle ø. These trigonometric functions are defined as follows:
Sine (sin ø): sin ø = y / r
Cosine (cos ø): cos ø = x / r
Tangent (tan ø): tan ø = y / x
Hence, the values of r, x, and y are related to the sine, cosine, and tangent of angle ø through the above equations.
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ABDE is an isosceles trapezoid. Select all parts that are marked congruent.
Side AB and Side AE
Angle E and Angle D
Angle E and Angle B
Angle A and Angle B
Angle A and Angle D
Side AE and Side BD
Side AB and Side ED
The concurrent parts in the given ABDE are Side AB ≅ Side ED and Side AE ≅ Side BD.
Concurrent triangles:Two triangles are said to be congruent if all three corresponding sides and all three corresponding angles have the same size. These triangles may be moved, flipped, twisted, and turned to get a similar effect. When relocated, they are parallel to one another. The concurrence sign is '≅'.
Here we have,
ABDE is an isosceles trapezoid.
Draw 2 diagonals in a given trapezoid
Then the 2 diagonal will divide the trapezoid into 4 triangles
Here ΔAOB, ΔBOD, ΔEOD, and AOE
From the formed triangles
ΔAOB and ΔEOD will be two concurrent triangles
Then Side AB ≅ Side ED
ΔBOD and AOE will be two concurrent triangles
Side AE ≅ Side BD
Therefore,
The concurrent parts in the given ABDE are Side AB ≅ Side ED and Side AE ≅ Side BD.
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Help me with geometry
Answer:
34
Step-by-step explanation:
This is somewhat hard to explain for me. I attached a picture/diagram to help.
They are congruent angles.
Angle N, inside the MNP triangle is 34. Which means the adjacent angle is 56.
The interior angles of a triangle add up to 180.
So to find angle M in triangle LNM, we can do 180 - 90 - 56.
Therefore the m<LMN = 34
Suppose that you’d like to retire in 40 years and you want to have a future value of $ 500000 in a savings account. Also suppose that your employer makes regular monthly payments into your retirement account.
If you can expect an APR of 7% for your account, how much do you need your employer to deposit each month?
Employer Contribution =
The formulas we have been using assume that the interest rate is constant over the period in question. Over a period of 40 years, though, interest rates can vary widely. To see what difference the interest rate can make, let’s assume a constant APR of 4% for your retirement account. How much do you need your employer to deposit each month under this assumption?
Employer Contribution =
a) If only your employer makes the monthly deposits into your retirement account for the future value of $500,000 in 40 years at 7% APR, your employer contributions should be $190.49.
b) If only your employer makes the monthly deposits into your retirement account for the future value of $500,000 in 40 years at 4% APR, your employer contributions should be $423.03.
What is the future value?The future value represents the amount expected to be the balance in the investment account after a period.
The future value can be used to determine the periodic payments required to meet the target balance, using an online finance calculator.
Periodic Deposits at 7%:N (# of periods) = 480 months (40 years x 12)
I/Y (Interest per year) = 7%
PV (Present Value) = $0
FV (Future Value) = $500,000
Results:
PMT (Periodic deposits) = $190.49
Sum of all periodic payments = $91,435.07
Total Interest = $408,564.93
Periodic Deposits at 4%:N (# of periods) = 480 (40 years x 12)
I/Y (Interest per year) = 4%
PV (Present Value) = $0
FV (Future Value) = $500,000
Results:
PMT (Periodic deposits) = $423.03
Sum of all periodic payments = $203,052.33
Total Interest = $296,947.67
Thus, the periodic deposits to meet the future value will depend on the interest rate.
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5.6.8 let f be a uniformly continuous function on a set e. show that if {xn} is a cauchy sequence in e then {f(xn)} is a cauchy sequence in f(e). show that this need not be true if f is continuous but not uniformly continuous.
Proof. (1): To prove {f(xn)} is a Cauchy sequence just need to prove ∀ > 0, ∃N, s.t.,∀n, m > N,
have |f(xn) − f(xm)| < . Since f is uniformly continuous on set E, thus ∀ > 0, ∃δ > 0, s.t.,
∀x, y ∈ E, if |x − y| < δ,then |f(x) − f(y)| < .as {xn} is a Cauchy sequence, then ∃N, s.t.,
∀n, m > N, |xn − xm| < δ, thus |f(xn) − f(xm)| < which proves {f(xn)} is a Cauchy sequence.
(2): for example f(x) = 1
x
, x ∈ (0, 2) which is continuous but not uniformly continuous. {
1
n
} is
a Cauchy sequence, however, {f(xn)} does not converge which proves that it is not a Cauchy
sequence.
Showing that this need not be true if f is continuous but not uniformly continuous.
Given :
let f be a uniformly continuous function on a set e. show that if {xn} is a cauchy sequence in e then {f(xn)} is a cauchy sequence in f(e).
( 1 )
If f is uniformly continuous on E, then given ε > 0 there is δ > 0 such that if x, y are in E and |x−y| < δ,
then |f(x) − f(y)| < ε. Let (xn) be a Cauchy sequence in E. Then given δ > 0 there is N such that if p, q > N,
then |xp − xq| < δ, and thus |f(xp) − f(xq)| < ε, implying that (f(xn)) is a Cauchy sequence.
( 2 )
Let E = {1, 1/2, 1/3, · · } and f(1/n) = 1 is n is odd, f(1/n) = −1 if
n is even. Then f is continuous but not uniformly continuous. The sequence (xn) = (1/n) in E is Cauchy but the
sequence (f(xn)) = (1, −1, 1, −1, · · ·) is not Cauchy.
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