Answer:
The value of the baseball increase $0.15
Step-by-step explanation:
we have
The expression above represent a linear equation in slope intercept form
where
The slope or unit rate is equal to
The y-intercept is equal to ----> original value of a baseball card
therefore
The value of the baseball increase $0.15 per year
Final answer:
The value of the baseball card is increasing by $0.15 per year. For the bat and ball problem, the ball costs $0.05 and the bat costs $1.05.
Explanation:
To find out how much the baseball card's value is increasing per year, we look at the formula given for the value of the baseball card: 0.15y + 0.35. In this formula, y represents the number of years since the card was released. The coefficient of y is 0.15, which indicates the yearly increase in the value of the card.
Therefore, the value of the baseball card is increasing by $0.15 per year.
Now let's look at the classic bat and ball problem. The total cost of the bat and the ball is $1.10, and the bat costs $1.00 more than the ball. If we denote the cost of the ball as x, then the cost of the bat is x + $1.00. Setting up the equation x + (x + $1.00) = $1.10, we combine like terms to get 2x + $1.00 = $1.10. Solving for x, we subtract $1.00 from both sides to get 2x = $0.10. Dividing both sides by 2 yields x = $0.05. So, the ball costs $0.05, and the bat costs $1.05.
The number of bocks has 9 in the ones place. The number in the hundreds place is one more then the tens place. Those two numbers equal 11. How many block are there?
There are 659 blocks
Solution:
The number in ones place is 9
Let's denote number in the tens place with x
The number in the hundreds place is one more then the tens place
Therefore,
The number in hundreds place is 1 + x
Those two numbers equal 11
Which means tens place number and hundreds place number sums up to 11
tens place + hundreds place = 11
x + 1 + x = 11
2x = 11 - 1
2x = 10
x = 5
Thus number in tens place = x = 5
Number in hundreds place = 1 + x = 1 + 5 = 6
The number is represented as:
Number = (Number in hundreds place)(number in tens place)(number in ones place)
Number = 659
Thus 659 blocks are there
You are assessing the one minute APGAR score for a newborn. She is pink all over and has a pulse of 130. As you dry her off she begins to cry vigorously and kick her legs. Her APGAR score is________________
Answer:
2.
Step-by-step explanation:
Healthy baby.
A gardener is planting two types of trees: Type A is 5 feet tall and grows at a rate of 13 inches per year. Type B is 3 feet tall and grows at a rate of 21 inches per year. Algebraically determine exactly how many years it will take for these trees to be the same height.
Answer:it will take 0.25 years
Step-by-step explanation:
Let x represent the number of years that it will take for type A and tree B to be the same height.
Let y represent the height of type A after x years.
Let z represent the height of type B after x years.
Type A is 5 feet tall and grows at a rate of 13 inches per year. This means that
y = 5 + 13x
Type B is 3 feet tall and grows at a rate of 21 inches per year. This means that
z = 3 + 21x
To determine the number of years it will take both trees to be of the same height, we would equate equation y to z. It becomes
5 + 13x = 3 + 21x
21x - 13x = 5 - 3
8x = 2
x = 2/8 = 0.25 years
Chase purchased a sweatshirt from the clearance rack. The price of the sweatshirt after the discount is represented by the expression 0.75x,where x represents the original price of the sweatshirt
Question is Incomplete; Complete question is given below;
Chase purchased a sweatshirt from the clearance rack. The price of the sweatshirt after the discount is represented by the expression 0.75x, where x represents the original price of the sweatshirt.
Which expression also represents the discounted price of the sweatshirt?
A (0.75 − 0.25)x
B (0.75 + 0.25)x
C x – 0.25x
D x + 0.25x
Answer:
C . [tex]x-0.25x[/tex]
Step-by-step explanation:
Given:
Expression representing price of the sweatshirt after the discount = [tex]0.75x[/tex]
[tex]x[/tex] ⇒ original price of sweatshirt
We need to find the expression which also represents the discounted price of the sweatshirt.
Solution:
Form the given expression we can see that after discount we are paying only 75% of the original amount of sweatshirt.
So we can say that;
The discount price was 25% of the original price i.e [tex]0.25x[/tex]
So now we can say that;
Price after discount is equal to difference of original price and discounted price.
framing in equation form we get;
Price after discount = [tex]x-0.25x[/tex]
Hence the equivalent expression for the given discounted price of sweatshirt is [tex]x-0.25x[/tex].
Answer:
Step-by-step explanation:
How many syllbals can you rap? The average is 8.8 syllbals per second. Mark Lee can rap 90.0 sps, also Rap mon can rap 45.5 sps. And finally Suga with 11.2 sps. So out these brave three who is the fastest?
Answer:
August D is the fastest rapper. XD
Step-by-step explanation:
Answer:
Based on this info I would say that Mark Lee does because he is at 90.0 sps.
Step-by-step explanation:
In an opinion poll, 25% of a random sample of 200 people said that they were strongly opposed to having a state lottery. The standard error of the sample proportion is approximately_________.
Answer: 0.031 .
Step-by-step explanation:
The standard error of the sample proportion is given by :-
[tex]SE_p=\sqrt{\dfrac{p(1-p)}{n}}[/tex]
, where p= Sample proportion and n is the sample size.
As per given , we have
In an opinion poll, 25% of a random sample of 200 people said that they were strongly opposed to having a state lottery.
i.e. p= 0.25 and n= 200
Then , the standard error of the sample proportion [tex]=\sqrt{\dfrac{0.25(1-0.25)}{200}}[/tex]
[tex]=\sqrt{\dfrac{0.25\times0.75}{200}}=\sqrt{0.0009375}\\\\=0.0306186217848\approx0.031[/tex]
Hence, the standard error of the sample proportion is approximately 0.031 .
Final answer:
The standard error of the sample proportion is approximately 0.0306.
Explanation:
The standard error of a sample proportion can be calculated using the formula:
Standard Error = √((p)(1-p)/n)
where p is the proportion of the sample and n is the sample size. In this case, the proportion is 0.25 (since 25% of the sample is strongly opposed to having a state lottery) and the sample size is 200. Plugging these values into the formula:
Standard Error = √((0.25)(1-0.25)/200) = √(0.1875/200) = √(0.0009375) = 0.0306
So, the standard error of the sample proportion is approximately 0.0306.
A guy wire is used to support flagpoles in windy areas. These wires are anchored in the ground to provide constant stability. You need to replace the guy wire on a pole that is 20 feet tall.. The anchor point for this flagpole is 15 feet away from the base of the pole. a. How long is the wire you need to purchase to replace the old wire? b. You are also helping a friend determine the length of a guy wire for a 10-foot flagpole. The wire needs to be anchored 6 feet from the base of the pole. You can purchase guy wires in whole foot lengths only. Using your knowledge of perfect squares, estimate the length of wire you would need to complete the project. How much wire would be left over?
Answer: a) Length of wire to replace old one=9ft
b) Length of wire=3.09ft
Leftover wire=0.91ft
Step-by-step explanation: from the right angle triangle, Tan&=20/15
&=53.13°
Length of wire,x=cos 31.13=x/15
15cos31.13
X=9
b) Tan&=10/6
Tan^-110/6
&=59.05
Length= 6cos59.04
X=3.09
4-3.09=.91= leftover wire
Using the Pythagorean theorem, the length of the guy wire needed for a 20-foot flagpole with a 15-foot distant anchor point is 25 feet, and for a 10-foot flagpole with a 6-foot distant anchor point, a 12-foot guy wire is needed, leaving approximately 4 inches of wire leftover.
The question involves applying the Pythagorean theorem to determine the lengths of guy wires needed to support flagpoles. For the first scenario, a 20-foot tall flagpole with a 15-foot distant anchor point, the guy wire's length can be found using the equation a^2 + b^2 = c^2, where 'c' is the hypotenuse (the guy wire), 'a' is one leg (the height of the flagpole which is 20 feet), and 'b' is the other leg (the distance from the base of the pole to the anchor point which is 15 feet). After calculation, the guy wire's length is found to be 25 feet.
For the second scenario with a 10-foot tall flagpole and a 6-foot distant anchor point, using the same theorem gives us the equation 10^2 + 6^2 = c^2. Estimating using perfect squares, we get c as the square root of 136, which is approximately 11.66 feet. Since guy wires can only be purchased in whole foot lengths, we would purchase a 12-foot guy wire, leaving approximately 0.34 feet (about 4 inches) of wire leftover.
Stephen & Richard share a lottery win of £2950 in the ratio 2 : 3. Stephen then shares his part between himself, his wife & their son in the ratio 3 : 5 : 2. How much more does his wife get over their son?
Answer:
Stephen's wife got £354 more than his son.
Step-by-step explanation:
Given:
Amount of Lottery = £2950
Now Given:
Stephen & Richard share a lottery amount in the ratio 2 : 3
Let the common factor between them be 'x'.
So we can say that;
[tex]2x+3x=2950\\\\5x = 2950[/tex]
Dividing both side by 5 we get;
[tex]\frac{5x}{5}=\frac{2950}{5}\\\\x = 590[/tex]
So we can say that;
Stephen share would be = [tex]2x =2\times 590 = \£1180[/tex]
Now Given:
Stephen then shares his part between himself, his wife & their son in the ratio 3 : 5 : 2.
Let the common factor between them be 'y'.
So we can say that;
[tex]3y+5y+2y=1180\\\\10y=1180[/tex]
Dividing both side by 10 we get;
[tex]\frac{10y}{10}=\frac{1180}{10}\\\\y=118[/tex]
So Stephen's wife share = [tex]5y = 5\times 118= \£590[/tex]
And Stephen's son share = [tex]2y=2\times118 =\£236[/tex]
Now we need to find how much more her wife got then her son.
To find how much more her wife got than her son we will subtract Stephen's son share from Stephen's wife share.
framing in equation form we get;
Amount more her wife got than her son = [tex]590-236 = \£354[/tex]
Hence Stephen's wife got £354 more than his son.
Suppose the quantity demanded, q, of a product when the price is p dollars is given by the equation p = 462 − 5 q , and the quantity supplied is given by the equation p = 2 q . Find the equilibrium price and quantity.
If the quantity demanded, q, of a product when the price is p dollars is given by the equation p = 462 − 5q, the equilibrium price and quantity are 132 dollars and 66.
An equation is a combination of expressions in a meaningful manner in terms of variables.
The given equation between the quantity demanded, q, of a product and the price, p dollars, is as follows:
p = 462 - 5q ...(1)
The quantity supplied is given by the equation as follows:
p = 2q ...(2)
Substitute p = 2q in the equation (1),
2q = 462 - 5q
2q + 5q = 462
7q = 462
[tex]q = \dfrac{462}{7}[/tex]
q = 66
The equilibrium quantity is 66.
Put q = 66 in equation (2),
p = 2 × 66
p = 132
The equilibrium price is 132 dollars.
Thus, the equilibrium price and quantity are 132 dollars and 66.
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The equilibrium quantity is 66 units, and the equilibrium price is $168.
Explanation:To find out the equilibrium price and quantity, we must make the quantity demanded equal to the quantity supplied. This essentially means that we need to solve the system of equations defined by the two product price functions therefore set 462 - 5q = 2q.
The next step is grouping the 'q' terms on one side and the constants on the other which gives us: 7q = 462. Dividing by 7 on both sides, we get q = 66, which is the equilibrium quantity.
Finally, by substituting q = 66 into either of the original equations (but be consistent, for this case I'll use the first equation: p = 462 - 5q), we find out that the equilibrium price is p = 462 - 5*66 = $168.
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A football field is sloped from the center toward the sides for drainage. The height h, in feet, of the field x feetfrom, the side, is given by h=-0. 00025x2 + 0 04x. Find the height of the field a distance of 35 feet from the side The height of the field is □ feet. Round to the nearest tenth as needed.
To find the height of the football field at a distance of 35 feet from the side, you plug x=35 into the equation h=-0.00025x^2 + 0.04x, resulting in a height of 1.1 feet after rounding to the nearest tenth.
Explanation:To find the height of the football field a distance of 35 feet from the side, we will substitute the value of x into the given quadratic equation h = -0.00025x2 + 0.04x. We can follow these steps:
First, plug in the value of x, which is 35, into the equation: h = -0.00025(35)2 + 0.04(35).Next, calculate the square of 35, which is 1225, and then multiply by -0.00025 to get -0.30625.Then, multiply 0.04 by 35 to get 1.4.Now, add the two results to get the height: h = -0.30625 + 1.4.After performing the addition, the height is h = 1.09375 feet.Lastly, round the result to the nearest tenth: the height of the field is 1.1 feet at a distance of 35 feet from the side.Suppose the radius of the sphere is increasing at a constant rate of 0.3 centimeters per second. At the moment when the radius is 24 centimeters, the volume is increasing at a rate of?
Step-by-step explanation:
We have equation for volume of a sphere
[tex]V=\frac{4}{3}\pi r^3[/tex]
where r is the radius
Differentiating with respect to time,
[tex]\frac{dV}{dt}=\frac{d}{dt}\left (\frac{4}{3}\pi r^3 \right )\\\\\frac{dV}{dt}=\frac{4}{3}\pi \times 3r^2\times \frac{dr}{dt}\\\\\frac{dV}{dt}=4\pi r^2\times \frac{dr}{dt}[/tex]
Given that
Radius, r = 24 cm
[tex]\frac{dr}{dt}=0.3cm/s[/tex]
Substituting
[tex]\frac{dV}{dt}=4\pi r^2\times \frac{dr}{dt}\\\\\frac{dV}{dt}=4\pi \times 24^2\times 0.3\\\\\frac{dV}{dt}=2171.47cm^3/min[/tex]
At the moment when the radius is 24 centimeters, the volume is increasing at a rate of 2171.47 cm³/min.
Points X and Z are on a number line, and point Y partitions XZ into two parts so that the ratio of the length of XY to the length of YZ is 5:7. The coordinate of X is 1.3, and the coordinate of Y is 3.8. What is the coordinate of Z?
Answer:
The coordinate of [tex]Z=7.3[/tex].
Step-by-step explanation:
To determine:
What is the coordinate of Z?
Fetching Information and Solution Steps:
Points X and Y are on a number linepoint Y partitions [tex]XZ[/tex] into two parts so that the ratio of the length of XY to the length of [tex]YZ[/tex] is 5:7The coordinate of X is 1.3the coordinate of Y is 3.8The distance [tex]XY[/tex] = [tex]3.8-1.3 = 2.5[/tex]
As
Y is [tex]\frac{5}{12}[/tex] of the way from [tex]X[/tex] to [tex]Z[/tex]
So,
[tex]YZ[/tex] is [tex]\frac{7}{5}[/tex] as big as [tex]XY[/tex]
Therefore,
[tex]Z = 3.8 + (\frac{7}{5} )(2.5)[/tex]
[tex]Z=3.8+\frac{17.5}{5}[/tex]
[tex]Z=7.3[/tex]
Therefore, the coordinate of [tex]Z=7.3[/tex].
Keywords: coordinate, point, length, ratio
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The coordinate of Z is (4.43, 11.57)
The coordinate of X is (1, 3)
The coordinate of Y is (3, 8)
Let the coordinate of z be (x₂, y₂)
The ratio of division is 5:7
m = 5, and n = 7
Let the coordinate of y be (a, b)
a = 3, b = 8
[tex]a=\frac{mx_1+nx_2}{m+n}\\\\3= \frac{5(1)+7x_2}{5+7}\\\\3(12)=5+7x_2\\\\7x_2=36-5\\\\7x_2=31\\\\x_2=4.43[/tex]
[tex]ab=\frac{my_1+ny_2}{m+n}\\\\8= \frac{5(3)+7y_2}{5+7}\\\\8(12)=15+7y_2\\\\7y_2=96-15\\\\7y_2=81\\\\y_2=11.57[/tex]
The coordinate of Z is (4.43, 11.57)
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Which would be the best trend line for the given data set?
A. y=-3/2x+8
B. y=3/2x+5
C. y=2/3x+8
D. y=-2/3x+5
They sold 64 adult tickets (x) and 132 kid tickets (y) and made $1040. An adult ticket is double the cost of a kid's ticket. How much is an adult ticket? How much is a kid's ticket? Here are my equations: x=2y 64x+132y=1040
Answer: the cost of one adult ticket is $8
the cost of one kid ticket is $4
Step-by-step explanation:
Let x represent the cost of one adult ticket.
Let y represent the cost of one kid ticket.
They sold 64 adult tickets (x) and 132 kid tickets (y) and made $1040. This means that
64x + 132y = 1040 - - - - - - - - - - -1
An adult ticket is double the cost of a kid's ticket. This means that
x = 2y
Substituting x = 2y into equation 1, it becomes
64 × 2y + 132y = 1040
128y + 132y = 1040
260y = 1040
y = 1040/260 = 4
x = 2y = 2 × 4
x = 8
John is building a custom wall for a kitchen remodel. He knows the space for the wall is square with an area of 121 feet. How long does his diagonal brace need to be?
Answer:
His diagonal brace need to be 15.55 feet long in the kitchen.
Step-by-step explanation:
Given : The wall of the kitchen is in the shape of a square.
Area of the wall = 121 sq ft
Let us assume the each side of wall = a ft
Now, AREA OF THE SQUARE = Side x Side
⇒ a x a = 121 sq units
[tex]\implies a ^2 = 121\\\implies a = 11[/tex]
So, each side of the kitchen wall is 11 ft.
Now,as we know in a square : EACH ANGLE IN THE SQUIRE IS A RIGHT ANGLE.
So, by Pythagoras Theorem, Diagonal D is:
[tex]D^2 = a^2 + a^2\\\implies D^2 = 11^2 + 11^2 = 121 + 121\\\implies D^2 = 242\\\implies D = 15.55[/tex]
Hence, his diagonal brace need to be 15.55 ft long in the kitchen.
You have a coupon worth $18 off the purchase of a scientific calculator.The calculator is also being offered at a discount of 15% off. You can only use either the coupon or the discount, but not both.Write a linear equation to represent using the coupon.Use x for the cost of the calculator And y for the cost after the coupon.
Answer:
The linear equation representing cost after using coupon is [tex]y=x-18[/tex].
Step-by-step explanation:
Given:
Value of coupon = $18
Let the cost of the calculator be 'x'.
And the Cost after redeeming the coupon be 'y'.
We need to write a linear equation to represent using the coupon.
Solution:
Now we can say that;
Cost after redeeming the coupon will be equal to cost of the calculator minus Value of coupon.
framing in equation form we get;
[tex]y=x-18[/tex]
Hence The linear equation representing cost after using coupon is [tex]y=x-18[/tex].
A swimmer ascended in the pool 2/3 meters at a time. She did this 8 times to reach the surface of the pool. What is the distance that represents the swimmer's total ascension
Answer:
Step-by-step explanation:
So 8 times more than 2/3
The total ascension of the swimmer can be calculated by multiplying the distance ascended each time by the number of ascents, resulting in 5 and 1/3 meters.
Explanation:The total ascension of the swimmer can be calculated by multiplying the distance ascended each time by the number of ascents. In this case, the swimmer ascended 2/3 meters 8 times.
Total ascension = 2/3 meters * 8 ascents = 16/3 meters = 5 and 1/3 meters.
Therefore, the distance representing the swimmer's total ascension is 5 and 1/3 meters.
Significant Figures: The last page of a book is numbered 764. The book is 3.0 cm thick, not including its covers. What is the average thickness (in centimeters) of a page in the book, rounded to the proper number of significant figures?
Answer:
0.0079 cm.
Step-by-step explanation:
We have been given that the last page of a book is numbered 764. The book is 3.0 cm thick, not including its covers.
We know that each page is marked on both sides, so we will find total number of pages by dividing 764 by 2 as:
[tex]\text{Total number of pages}=\frac{764}{2}[/tex]
[tex]\text{Total number of pages}=382[/tex]
To find the average thickness of each page, we will divide thickness of book by total number of pages as:
[tex]\text{Average thickness of each page}=\frac{3.0}{382}[/tex]
[tex]\text{Average thickness of each page}=0.0078534031413613[/tex]
We can see that there are 3 significant figures in 764 and 2 significant digits in 3.0.
We know that the result of a multiplication or division is rounded to the number of significant figures equal to the smallest number of significant figures among the numbers being multiplied/divided. So we need to round our answer to 2 significant digits.
[tex]\text{Average thickness of each page}\approx 0.0079[/tex]
Therefore, the average thickness of each page is approximately 0.0079 cm.
A principal placed a total of 288 math books on 8 book charts. He placed an equal number of books on each cart. Each book is 6 pounds. How much pounds of books was placed on the last book cart.
Final answer:
The total weight of books on the last book cart is 216 pounds, found by dividing the total number of books by the number of carts and then multiplying by the weight per book.
Explanation:
The question involves dividing the total number of math books evenly among book carts and then determining the total weight of books on one cart. Here's a step-by-step solution:
First, divide the total number of books, which is 288, by the number of book carts, which is 8, to find the number of books per cart: 288 \/ 8 = 36 books per cart.
Next, since each book weighs 6 pounds, multiply the number of books per cart by the weight per book to get the total weight on one book cart: 36 books x 6 pounds/book = 216 pounds.
Therefore, the total weight of books on the last book cart is 216 pounds.
Perform the requested operation or operations.
f(x) = 4x + 9, g(x) = 4x2
Find (f + g)(x).
16x3 + 36x
four x plus nine divided by four x squared.
4x + 9 - 4x2
4x + 9 + 4x2
Answer:
[tex]4x+9+4x^2[/tex]
Step-by-step explanation:
The given functions are:
[tex]f(x)=4x+9[/tex] and [tex]g(x)=4x^2[/tex]
We want to find [tex](f+g)(x)[/tex].
Recall that:
[tex](f+g)(x)=f(x)+g(x)[/tex]
We substitute the functions to get:
[tex](f+g)(x)=4x+9+4x^2[/tex]
We now write in standard form to obtain:
[tex](f+g)(x)=4x^2+4x+9[/tex]
Two planes flying opposite directions (north and south) pass each other 80 miles apart at the same altitude. The northbound plane is flying 300 mph (miles per hour) and the southbound plane is flying 150 mph. How far apart are the planes in 20 minutes? (Round your answer to one decimal place.) mi When are the planes 500 miles apart? (Round your answer to one decimal place.)
Answer:
a). 170 miles
b). Planes will be 500 miles apart after 1.1 hours.
Step-by-step explanation:
a). Two planes heading towards North and South when passed each other were 80 miles apart.
Distance between these planes after 20 minutes or [tex]\frac{1}{3}[/tex] hours.
From the figure attached,
In right angle triangle DEC,
DE = 80 miles
BE = (Speed × duration) = [tex]300\times \frac{1}{3}=100[/tex] miles
Similarly, BC = [tex]150\times \frac{1}{3}=50[/tex] miles
By Pythagoras theorem,
DC² = EC² + DE²
= (EB + BC)² + DE²
= (100 + 50)² + (80)²
= 28900
DC = √28900 = 170 miles
b). Now we have to evaluate the duration after which distance between the planes is 500 miles.
Let after t hours planes will be 500 miles apart.
Then EB = 300t
BC = 150t
Therefore, EC = EB + BC = 450t
It's given that DC = 500 miles
By Pythagoras theorem again,
DC² = EC²+ DE²
(500)²= (450t)²+ (80)²
250000 = 202500t² + 6400
2500 = 2025t² + 64
2025t² = 2436
t² = 1.20297
t = 1.097 hours ≈ 1.1 hours
Therefore, both the planes will be 500 miles apart after 1.1 hours.
Carole needs 4pounds of nuts for her granola. She has 26ounces of walnuts and 28ounces of cashews. How many ounces of peanuts should she buy so she has 4pounds of nuts?
Answer:
10 oz
Step-by-step explanation:
there are 16 oz in a pound. 4 * 16 = 64 so she needs 64 oz in all. she has 26 oz of walnuts and 28 oz of cashews. add these together and subtract it from 64 to find how many oz of peanuts she needs.
26 + 28 = 54
64 - 54 = 10
she needs 10 oz of peanuts
3. Use the Division Algorithm to establish the following: (a) The square of any integer is either of the form 3k or 3k + 1. (b) Thecubeofanyintegerhasoneoftheforms:9k,9k+1,or9k+8. (c) The fourth power of any integer is either of the form 5k or 5k + 1.
Explanation
Lets use congruence
For the square of an integer lets use congruence module 3
If the congruence module 3 is 0, then the congruence of the sqaure is 0² = 0If the congruence is 1, then the congruence of the square is 1² = 1If the congruence is 2, then the congruence of the square is 2² = 4 = 4-3 = 1 (4 and 1 are equal is Z₃)Thus, the square of an integer has the form 3k (the congruence is 0) or 3k+1 (the congruence is 1).
For the cube, lets use congruence module 9
if the congruence module 9 is 0, then the congruence of the cube is 0³ = 0if the congruence is 1, then the congruence of the cube is 1³ = 1if the congruence is 2, then the congruence of the cube is 2³ = 8if the congruence is 3, then the congruence of the cube is 3³ = 27 = 0if the congruence is k+3, for certain k, then the congruence of the cube is (k+3)³ = k³+9k²+27k+9 = k³. Hence the congruence of the cube is the same after addding 3 to a number. Thus, the congruence of a number with congruence 4,5,6,7 or 8 module 9 is obtained by computing the congruence module 9 of 1,2 or 3.With the argument given above, we obtain that the congruence module 9 of the cube of a number is always 0,1 ir 8, thus, the cube of an integer has the form 9k, 9k+1 or 9k+8.
As for the fourth power, we take congruence module 5:
If the congruence module 5 is 0, then the congruence of the fourth power is 0⁴ = 0if the congruence is 1, then the congruence of the foruth power is 1⁴ = 1If the congruence is 2, then the congruence of the fourth power is 2⁴ = 16 = 16-5*3 = 1If the congruence is 3, then the congruence of the fourth power is 3⁴ = 81 = 81-5*16 = 1If the congruence is 4, then the congruence of the fourth power is 4⁴ = 256 = 256-5*51 = 1In any case, the congruence of the fourth power module 5 is either 0 or 1, as a result, the fourth power has the form 5k or 5k+1.
Final answer:
Using the Division Algorithm, we show that the square of an integer is in the form of 3k or 3k+1, the cube is in the form of 9k, 9k+1, or 9k+8, and the fourth power is in the form of 5k or 5k+1.
Explanation:
The Division Algorithm states that if we divide any integer n by a positive integer d, there are unique integers q and r such that n = dq + r and 0 ≤ r < d. Using this and thinking about integers as being of the form 3q, 3q + 1, or 3q + 2 we can prove the following statements:
(a) The square of any integer:
Any integer can be written as 3k, 3k + 1, or 3k + 2. Squaring these forms:
(3k)² = 9k² = 3(3k²)(3k + 1)² = 9k² + 6k + 1 = 3(3k² + 2k) + 1(3k + 2)² = 9k² + 12k + 4 = 3(3k² + 4k + 1) + 1In each case, the square of an integer is either of the form 3k or 3k + 1.
(b) The cube of any integer:
Cubing the three forms:
(3k)³ = 27k³ = 9(3k³)(3k + 1)³ = 27k³ + 27k² + 9k + 1 = 9(3k³ + 3k² + k) + 1(3k + 2)³ = 27k³ + 54k² + 36k + 8 = 9(3k³ + 6k² + 4k) + 8The cube is of the form 9k, 9k + 1, or 9k + 8.
(c) The fourth power of any integer:
For 3k and 3k + 1, the fourth powers are:
(3k)⁴ = 81k⁴ = 5(16k⁴)(3k + 1)⁴ = 81k⁴ + 108k³ + 54k² + 12k + 1 = 5(16k⁴ + 21k³ + 10k² + 2k) + 1For 3k + 2, because 2⁴ = 16 is already a multiple of 5, (3k + 2)⁴ will also have the form 5m or 5m + 1. Therefore, the fourth power of an integer is either of the form 5k or 5k + 1.
Mr. And Mrs. Kingsley went out for dinner on Valentine's Day. Their dinner bill came to $57.24. If they wish to leave a 15% tip how much will they pay in total
Answer:
65.83
Step-by-step explanation:
The total amount Mr. And Mrs. Kingsley have to leave if Their dinner bill came to $57.24, If they wish to leave, a 15% tip is $65.83.
What is percentage?As it is clear from the term per cent means measuring anything per hundred. For example, you get marks per 100 marks, so you get the percentage.
Given:
The dinner bill is $57.24,
The tip percentage = 15%,
Calculate the amount of the tip as shown below,
The amount of the tip = 15% of 57.24
The amount of the tip = 15 / 100 × 57.24
The amount of the tip = 0.15 × 57.24
The amount of the tip = 8.586
Hence, total amount = tip amount + bill amount
Total amount = 57.24 + 8.586
Total amount = 65.83
Thus, the total amount is $65.83.
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A rectangular tank that is 5324 ft cubed with a square base and open top is to be constructed of sheet steel of a given thickness. Find the dimensions of the tank with minimum weight.
Answer:
Side of 22 and height of 11
Step-by-step explanation:
Let s be the side of the square base and h be the height of the tank. Since the tank volume is restricted to 5324 ft cubed we have the following equation:
[tex]V = s^2h = 5324[/tex]
[tex]h = 5324 / s^2[/tex]
As the thickness is already defined, we can minimize the weight by minimizing the surface area of the tank
Base area with open top [tex]s^2[/tex]
Side area 4sh
Total surface area [tex]A = s^2 + 4sh[/tex]
We can substitute [tex]h = 5324 / s^2[/tex]
[tex]A = s^2 + 4s\frac{5324}{s^2}[/tex]
[tex]A = s^2 + 21296/s[/tex]
To find the minimum of this function, we can take the first derivative, and set it to 0
[tex]A' = 2s - 21296/s^2 = 0[/tex]
[tex]2s = 21296/s^2[/tex]
[tex]s^3 = 10648[/tex]
[tex]s = \sqrt[3]{10648} = 22[/tex]
[tex]h = 5324 / s^2 = 5324 / 22^2 = 11[/tex]
To find the dimensions of the rectangular tank with a square base and minimum weight, we set up the volume and surface area formulas in terms of the base's side length and the tank's height, then use calculus to find the dimensions that minimize the surface area.
Explanation:The problem is a typical unconstrained optimization problem in calculus. The volume of the tank is given as 5324 cubic feet. Since it's a rectangular tank with a square base, we can let x be the length of the side of the base and h be the height of the tank. Then, the volume of the tank is x^2*h = 5324.
The weight of the tank is proportional to the amount of steel used, which, in turn, is proportional to the surface area of the tank. The surface area of the tank is x^2 + 4*x*h. From the volume formula, we can express h in terms of x: h = 5324 / x^2. Substituting it to the surface area formula, we get the surface area is a function of x, such that "A = x^2 + 4x * 5324 / x^2".
To find the dimensions that minimize the weight, we find the derivative of A with respect to x and set it equal to 0. This will give the x that minimizes the surface area. Then substitute this value back into the equation h = 5324 / x^2 to get the optimal height.
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Find the sum . 1 1/3+(-2 1/6 ) =? Please write out the steps
Answer:
-5/6
Step-by-step explanation:
covert the mixed number into a improper fraction
4/3 + (- 13/6) =
Find LCD which is 6 so
[tex]\frac{4}{3} * 2[/tex]
8/6 + (-13/6) =
Which is -5/6
Consider a circle whose size can vary. The circumference of the circle is always 2π times as large as its radius. Let r represent the radius of the circle (in feet) and let C represent the circumference of the circle (in feet).
a. Write a formula that expresses C in terms of r.
b. Use your formula from part (a) to determine the circumference of a circle whose radius is 6 feet feet
c. Write a formula that expresses r in terms of C.
d. Use your formula from part (c) to determine the radius of a circle whose circumference is 37
Answer:a) C = 2πr
b) 37.68feet
c) r = C/2π
d) 5.89feet
Step-by-step explanation:
The answers are a) 2πr, b) 37.68 ft, c) r = C/2π and d) 5.8 ft
What is circumference?The circumference is the distance around the edge of a circle.
Given that, a circle whose size can vary. The circumference of the circle is always 2π times as large as its radius and r represent the radius of the circle (in feet) and let C represent the circumference of the circle (in feet).
a) C = 2πr
b) C = 2*3.14*6 = 37.68ft
c) C = 2πr, r = C/2π
d) r = 37/2*3.14 = 5.8 ft
Hence, The answers are a) 2πr, b) 37.68 ft, c) r = C/2π and d) 5.8 ft
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The process of using variables to represent unknown quantities and then finding relationships that involve these variables is referred to as ____________.
Answer:
Mathematical modeling
Step-by-step explanation:
Mathematical modeling is defined as translating the problems from an application area using the mathematical formulas.
The numerical analysis and theoretical analysis gives an insight and answers or guidance which is useful for the originating application.
It provides with the precision and direction for the solution of the problem.
The relationship can be modeled by the quadratic equation p = − 100s^2 + 2400s − 8000, where p represents the profit and s represents the selling price. Which selling price will maximize profits? A.$12 B.$20 C.$8 D.$4
Answer:
A. $12
Step-by-step explanation:
For a quadratic of the form ax²+bx+c, the axis of symmetry is given by ...
x=-b/(2a)
For your quadratic function, the axis of symmetry is ...
s = -(2400)/(2(-100)) = 12
The function has its extreme value on the axis of symmetry. Here, the leading coefficient is negative, so the parabola opens downward. That means the extreme value is a maximum.
Profit will be a maximum at a selling price of $12.
A triangular prism is 19.1 meters long and has a triangular face with a base of 18 meters and a height of 12 meters. The other two sides of the triangle are each 15 meters. What is the surface area of the triangular prism?
Answer:
A= 1132.8 m^2
Step-by-step explanation:
l=19.1m
b=18m
h=12m
s=15m
The formula to find surface area of the triangular prism is:
A = bh + 2ls + lb
A = (18×12) + 2(19.1×15) + (19.1×18)
A = 216 + 573 + 343.8
A= 1132.8 m^2
Answer: surface Area of triangular prism = 502.5m^2
Step-by-step explanation:
Formular for surface area of a prism is given as: SA=bh+(S1 +S2+S3)H
Given base =18m, length=19.1, h=12, w=15
S1=1/2bh =1/2×18×12 =108m
S2=108m . This is because the dimensions of the two ends of the triangular prism.
S3=lw=19.1×15=286.5m
Surface area=108+108+286.5=502.5m^2