What is the area in square graph units, of the triangle?
Answers
To solve the question, we will need to get the dimensions of the triangle, to do this, we will re-sketch the image as follow:
If we count the units of squares in the image, we will discover that the dimensions are as shown above
where
[tex]\begin{gathered} AE=4 \\ AC=\sqrt[]{8^2+10^2}=\sqrt[]{164}=12.806 \\ ED=\sqrt[]{6^2+8^2}=\sqrt[]{100}=10 \end{gathered}[/tex]Hence, we will use heron's formula to solve for the area
The formula to be used is
[tex]A=\sqrt[]{s(s-a)(s-b)(s-c)}[/tex]where
[tex]undefined[/tex]
Related Questions
Complete the table to find the derivative of the function.
Answers
Given the following original function:
[tex]\text{ y = }\frac{\text{ 4}}{9x^4}[/tex]Rewriting it will be:
[tex]\text{ }\frac{\text{ 4}}{\text{ 9}}x^{-4}[/tex]Differentiating it will be:
[tex]\frac{4(-4)}{9}x^{-5}[/tex]Simplifying it will be:
[tex]\text{ -}\frac{16}{9x^5}[/tex]7. A drag racer accelerated from 0m/s to 200 m/s. what is the acceleration?
Answers
Let's begin by listing out the information given to us:
Initial speed = 0 m/s
Final speed = 200 m/s
[tex]undefined[/tex]How do I find the volume of a cylinder with a diameter of 20 cm and a height of 15 cm in terms of Pi?
Answers
Cylilnder volume formula
[tex]V=\pi\frac{D^2}{4}h[/tex]where D is the diameter and h is the height of the cylinder.
Substituting with D = 20 cm, and h = 15 cm, we get:
[tex]\begin{gathered} V=\pi\cdot\frac{20^2}{4}\cdot15 \\ V=\pi\cdot\frac{400}{4}\cdot15 \\ V=1500\pi\text{ }cm^3 \end{gathered}[/tex]What is the result when the number 34 is decreased by 5%?
Answers
when 34 is decreased by 5%
Let's first find 5% of 34
5% of 34 =1. 7
34 decreased by 5% = 34 - 1.7 =32.3
Hamish has $25.97 in his pocket to spend at the craft store. He wants tobuy one paint canvas that cost $14 and paint pens that cost $3.15 each.How many paint pens can Hamish buy without going over his budget. *
Answers
We have the next information
he has $ 25.97
he wants to buy
one canvas that cost $14
and paint pens that cost $3.15 each
x is the number of paint pens
the inequality for this problem will be
[tex]14+3.15x\le25.97[/tex]the equal less than is that we can't buy over this quatity
we need to isolate the x in order to find how many paint pens Hamish can buy
[tex]\begin{gathered} 3.15x\le25.97-14 \\ 3.15x\le11.97 \\ x\le\frac{11.97}{3.15} \\ x\le3.8 \end{gathered}[/tex]Hamish can buy 3 paint pens
I need help with my math
Answers
Given
a = 10 in
c = 26 in
b = ?
Procedure
Pythagorean theorem, the well-known geometric theorem that the sum of the squares on the legs of a right triangle is equal to the square on the hypotenuse (the side opposite the right angle)—or, in familiar algebraic notation, a2 + b2 = c2.
[tex]\begin{gathered} b=\sqrt[]{c^2-a^2} \\ b=\sqrt[]{26^2-10^2} \\ b=\sqrt[]{676-100} \\ b=\sqrt[]{576} \\ b=24 \end{gathered}[/tex]The answer would be b = 24.
If the unit rate is 60 miles per hour, how many miles would be driven in 5 hours?
Answers
A rate of 60 miles per hour indicates that every hour a mile is driven. If you wnat to know how many miles are driven in 5 hours then you must multiply the rate by the number of hours driven:
[tex]60\text{ }\frac{\text{mi}}{\text{h}}\cdot5\text{ h}=300\text{ mi}[/tex]AnswerThen the answer is 300 miles.
The Senate in a certain state is comprised of 57 Republicans, 40 Democrats, and 3 Independents. How many committees can be formed if each committee musthave 3 Republicans and 2 Democrats?How many committees can be formed?
Answers
Data
Number of republicans = 57
Number of democrats = 40
Independents = 3
Procedure
1.- Divide 57 by 3
57/3 = 19
2.- Divide 40 by 2
40/2 = 20
Well, it can be formed 19 commitees
Directions: Fill in the blank measurements1. (triangle) Base=32mm Height:35mmArea:___2.(Parallelogram)Area:168yd^2 Height:8ydBase:-----
Answers
Question 1
Given:
Base of triangle, b = 32 mm
Height of triangle, h = 35 mm
Let's find the area of the triangle.
To solve for the area, apply the formula:
[tex]\text{Area of triangle = }\frac{1}{2}\ast b\ast h[/tex]Thus, we have:
[tex]\begin{gathered} \text{Area = }\frac{1}{2}\ast32\ast35 \\ \\ \text{Area = 16}\ast35 \\ \\ \text{ Area = 560 mm}^2 \end{gathered}[/tex]Area: 560 mm²
• Question 2:
Given:
Area of parallelogram, A = 168 yd²
Height of parallelogram, h = 8 yd
Let's find the base, b, of the parallelogram.
To find the base, apply the area of a parallelogram formula:
[tex]\text{Area}=b\ast h[/tex]Input values into the formula:
[tex]168=b\ast8[/tex]Let's solve for b.
Divide both sides by 8:
[tex]\begin{gathered} \frac{168}{8}=\frac{b\ast8}{8} \\ \\ 21=b \\ \\ b=21 \end{gathered}[/tex]Therefore, the base of the parallelogram is 21 yards
ANSWER:
• 1. Area: ,560 mm,²
• 2. Base: ,21 yd
(b) The diagram below shows a quadrilateral with the length of its sides written in terms of x. 1.8 (15-2x) cm (3x - 7) cm (2x + 5) cm (i) Write an expression, in terms of x, for the perimeter of the quadrilateral. [2] Express your answer in its simplest form. The perimeter of the quadrilateral is 32 cm. Find the longest side of the quadrilateral. (ii) (2x - 1) cm [2]
Answers
Explanation
From the image, we will have the sides of the quadrilateral as:
[tex]\begin{gathered} 1)(15-2x)cm \\ 2)(3x-7)cm \\ 3)(2x-1)cm \\ 4)(2x+5)cm \end{gathered}[/tex]Part I
Therefore, the perimeter of the quadrilateral is the sum of all the sides. This is given as
[tex]\begin{gathered} p=15-2x+3x-7+2x-1+2x+5 \\ group\text{ like terms} \\ p=-2x+3x+2x+2x+15-7-1+5 \\ p=5x+12 \end{gathered}[/tex]Answer: (5x+12) cm
Part II
Since the perimeter of the quadrilateral is 32cm. Therefore,
[tex]\begin{gathered} 5x+12=32 \\ 5x=32-12 \\ 5x=20 \\ x=\frac{20}{5}=4cm \end{gathered}[/tex]From the image it is clear that the longest side is 2x+5, we can then convert this to its actual length value.
[tex]longest\text{ side =2x+5=}2(4)+5=8+5=13[/tex]Answer: 13cm
I'm looking for the answer to angle b if 36° is the other angle
Answers
to solve this question, we can use two methods
1. relate it with opposite angles and subtract the angles from 360
2. use angle on a straight line is equal to 180 degree
i'll use the second method simply because it's faster and saves time
[tex]\begin{gathered} 36^0+b^0=180^0 \\ \text{reason: angles on a straight line is equal to 180 degre}e \\ b=180-36 \\ b=144^0 \end{gathered}[/tex]from the calculation above, the value of angle b is 144 degrees
How many solutions does this equation have?8 – 7y = -6y
Answers
To know the number of solutions the equation have, we will follow the steps below:
8 - 7y = -6y
add 7y to both-side of the equation
8 - 7y+7y = -6y + 7y
8 = y
y=8
It has one solution
Therefore the equation has one solution
:
8 - 7y =
Tory creates a cell phone app. He wants to distribute it through a company that will pay him 100 plus 0.25 every time the app, x is downloaded. In the first he made 380. The equation that can be used to determine how many times the app, x, was downloaded is ____________
Answers
Given data:
The company will pay him 100 plus 0.25 every time the app(x): 100+0.25x
In the first he made 380
The equation that can be used to determine how many times the app, x, was downloaded is [tex]380=100+0.25x[/tex]Gianna is deciding between two landscaping companies for her place of business. Company A charges $50 per hour and a $150 equipment fee. Company B charges $25 per hour and a $250 equipment fee. Let AA represent the amount Company A would charge for tt hours of landscaping, and let BB represent the amount Company B would charge for tt hours of landscaping. Graph each function and determine the number hours, t,t, that would make the cost of each company the same.
Answers
We are given that company A charges $50 per hour and a fixed fee of $150. The total cost "Ca" is then given by the product of the number of hours "t" by the fee per hour plus the fixed fee. This is written mathematically as:
[tex]C_a=50t+150[/tex]Company B charges $25 per hour and a fixed fee of $250. If Cb is the cost for company B, then we have:
[tex]C_b=25t+250[/tex]To determine the number of hours "t" for when the cost is the same, then we need to set both costs equal:
[tex]C_a=C_b[/tex]Now we substitute the expression for each cost:
[tex]50t+150=25t+250[/tex]Now we solve for "t", first by subtracting "25t" from both sides:
[tex]\begin{gathered} 50t-25t+150=250 \\ 25t+150=250 \end{gathered}[/tex]Now we subtract 150 from both sides:
[tex]\begin{gathered} 25t=250-150 \\ 25t=100 \end{gathered}[/tex]Now we divide both sides by 25:
[tex]t=\frac{100}{25}=4[/tex]Therefore, the cost for both companies is the same after 4 hours.
To graph the functions we need to have into account that both equations represent lines since they are of the form:
[tex]y=mx+b[/tex]Where "m" is the slope and "b" is the y-intercept. Since the equations are lines we need to determine two points in each one of them.
Let's take the equation for compañy "a" and we replace the value t = 0, we get:
[tex]\begin{gathered} C_a=50(0)+150 \\ C_a=150 \end{gathered}[/tex]Therefore, the point (0, 150) is in the line for company "a". Now we substitute t = 1, we get:
[tex]\begin{gathered} C_a=50(1)+150 \\ C_a=200 \end{gathered}[/tex]Therefore, the point (1, 200) is also on the line. Now we plot the two points and join them with a line to get the graph. It looks like this:
Now we use the same procedure for company B and we get the following graph:
We notice that the interception point between the lines is the point where the costs are the same.
Solve. 0.8 + 0.3x = 5
Answers
We have the following equation:
[tex]0.8+0.3x=5[/tex]By subtracting 0.8 to both sides, we have
[tex]0.3x=4.2[/tex]and by dividing both sides by 0.3, we get
[tex]\begin{gathered} x=\frac{4.2}{0.3} \\ x=14 \end{gathered}[/tex]Therefore, the answer is: x= 14
35. A system of two linear equations in two unknowns where the lines are parallel with differing y-intercepts is said to bea. Dependent and inconsistentb. Independent and inconsistentc. Dependent and consistentd. Independent and consistente. None of the above
Answers
Step 1: Concept
When two equations have the same slope but different y-axis, they are parallel. Since there are no intersection points, the system has no solutions.
Step 2:
When you graph the equations, both equations represent the same line. If a system has no solution, it is said to be inconsistent.
Step 3:
Point1: a system of linear equations: A set of two or more equations made up of two or more variables that are considered simultaneously.
Point2: dependent system: A system of linear equations in which the two equations represent the same line; there are an infinite number of solutions to a dependent system.
Point3: inconsistent system: A system of linear equations with no common solution because they represent parallel lines, which have no point or line in common.
Point4: independent system: A system of linear equations with exactly one solution pair (x,y).
Final answer
None of the above
in 2003, Carlos Zambrano earned $340,000 pitching for the Chicago Cubs. In 2009 his salary was $18,750,000. a. Zambrano pitched in 32 games in 2003. b. He pitched 38 games in 2009. What was his salary per game in 2009. Round to the nearest cent. c. Did he earn more per game in 2009 than he did the entire 2003 season?
Answers
Answer:
a) His salary per game in 2003 was of $10,625.
b) His salary per game in 2009 was of $493,421.05.
c) Yes
Step-by-step explanation:
Salary per game:
How much Zambrano earned for each game he pitched. It is found dividing his salary by the number of games pitched.
a)
2003:
Earned $340,000.
Pitched 32 games.
340,000/32 = $10,625
His salary per game in 2003 was of $10,625.
b)
2009:
Earned $18,750,000.
Pitched 38 games.
18750000/38 = $493,421.05
His salary per game in 2009 was of $493,421.05.
c)
In the entire 2003 season, he earned $340,000.
Per game in 2009, he earned $493,421.05.
So yes, he earned more per game in 2009 than he did the entire 2003 season.
part a is correct please assist with part b c. determine the total enrollment of full time students in 2017 d. in what year is the total enrollment of full time students expected to be 2510?
Answers
b. t represents the number of years after 2014 and N represents the number of students enrolled.
"The product of the number of years after 2014 (t) and 110" is equivalent to 110t.
To calculate the number of students enrolled (N), we have to add 1850 to the previous amount, that is,
[tex]N=110t+1850[/tex]c. 2017 is 3 years after 2014, then we have to substitute t = 3 into the above equation, as follows:
[tex]\begin{gathered} N=110\cdot3+1850 \\ N=330+1850 \\ N=2180 \end{gathered}[/tex]The number of students enrolled in 2017 was 2180
d. Substituting N = 2510 into the equation and solving for t:
[tex]\begin{gathered} 2510=110t+1850 \\ 2510-1850=110t+1850-1850 \\ 660=110t \\ \frac{660}{110}=\frac{110t}{110} \\ 6=t \end{gathered}[/tex]6 years after 2014, that is, in 2020 it is expected that the number of students will be 2510
The table shows that Marta's heart beats 18 times every 15 s. Use equivalent ratios to complete the table. Explain how you found the time in seconds for 180 heartbeats.
Answers
Answer:
(a) 36 and 90
(b)150 times
Explanation:
In 15 seconds, Martha's heart beats 18 times.
Thus:
[tex]In\text{ 1 second, Martha's heart beats }\frac{18}{15}\text{ times}[/tex]Therefore:
[tex]\begin{gathered} In\text{ 30 seconds, Martha's heart beats }\frac{18}{15}\times30=36\text{ times} \\ In\text{ 45 seconds, Martha's heart beats }\frac{45}{15}\times30=90\; \text{times} \end{gathered}[/tex](b)If the number of heartbeats = 180
We use the ratio in the first row:
[tex]\begin{gathered} \frac{15}{x}=\frac{18}{180} \\ \frac{15}{x}=\frac{1}{10} \\ x=15\times10 \\ x=150\text{ seconds} \end{gathered}[/tex]Then, the number of seconds will be 150 seconds.
Find the value of constant k (or indicate that no such value of k exists), so that the average rate of change in F(t) = kt^2 on the interval [-1, 2] is equal to 1. K = ________
Answers
k = 1
Explanation:Given the function:
[tex]f(t)=kt^2[/tex]The average rate of change is given by the formula:
[tex]R=\frac{f(b)-f(a)}{b-a}[/tex]where a = -1, b = 2
[tex]\begin{gathered} R=\frac{2^2k-(-1)^2k}{2-(-1)} \\ \\ =\frac{4k-k}{2+1} \\ \\ =\frac{3k}{3}=k \end{gathered}[/tex]Now, the rate of change has been given to be 1, so
[tex]R=k=1[/tex]suppose the original quantity is $30 and the new quantity is $39. estimate the percent change. is this an example of a percent increase or a percent decrease?
Answers
original quantity = $30
New quantity = $39
increase amount = 39-30 = 9
Percent change:
30 x = 9
x = 9/30
x = 0.3
0.3x 100 = 30%
Since the quantity grows, it's a percent increase.
A drama club is planning a bus trip to New York City to see a Broadway play. The cost per person for the bus rental varies inversely to the number of people going on the trip. It will cost $30 per person if 44 people go on the trip. How much will it cost per person if 20 people go on the trip? Round your answer to the nearest cent, if necessary.
Answers
Statement Problem: A drama club is planning a bus trip to New York City to see a Broadway play. The cost per person for the bus rental varies inversely to the number of people going on the trip. It will cost $30 per person if 44 people go on the trip. How much will it cost per person if 20 people go on the trip?
Solution:
Let the cost per person for the bus rental be c;
Let the number of people going on the trip be n;
[tex]c\propto\frac{1}{n}[/tex][tex]\begin{gathered} c=\frac{k}{n} \\ \text{Where k is the constant of proportionality;} \\ k=cn \\ c=30,n=44 \\ k=30\times44 \\ k=1320 \end{gathered}[/tex]Then, we have;
[tex]\begin{gathered} \text{When;} \\ n=20,c=\text{?} \\ c=\frac{k}{n} \\ c=\frac{1320}{20} \\ c=66.00 \end{gathered}[/tex]If 20 people go on the trip, the cost per person is;
[tex]\text{ \$66.00}[/tex]if two _______ and included angle of one triangle are ______ to the corresponding two sides and included angle of other triangles ,then the _____ are congruent
Answers
if two _______ and included angle of one triangle are ______ to the corresponding two sides and included angle of other triangles ,then the _____ are congruent
ExplanationAccording to the statement given ,
The blanks will be fill in the blanks as follows.
If two sides and the included angle of one triangle are equal to the corresponding two sides and included angle of other triangle, then the triangles are congruent.
Answer
The bold and underline letters in the explanation part is the answer.
Identify a Ferris wheel with a radius of 9.5 m that rotates fully once every 10 seconds on a numeric model (minimum 5 data points).
Answers
Given:
The radius of the Ferries wheel is 9.5m.
The number of second the ferries wheel take to rotate fully =10 seconds.
Aim:
We need to find the numeric model for the given situation.
Explanation:
The full rotation is 360 degrees.
The wheel rotates 360 degrees per 10 seconds.
Divide 360 by 10, we get
[tex]\frac{360}{10}=36^{o\text{ }}\text{ per second.}[/tex]The wheel rotates 36 degrees per second.
Take that we traveled t seconds.
The point on the circle is y.
The angle is 36t degrees.
The height from the ground is 9.5+h.
The radius is 9.5m.
Consider the triangle that makes an angle of 36 t degrees,
Adjacent side = 9.5-h m and hypotenuse is 9.5 m.
Consider the cosine formula.
[tex]\cos \theta=\frac{\text{Adjacent side}}{\text{Hypotenuse}}[/tex]Substitute adjacent side = 9.5-h m and the hypotenuse is 9.5 m.
[tex]\cos 36t^o=\frac{9.5-h}{9.5}[/tex][tex]\cos 36t^o=\frac{9.5-h}{9.5}[/tex][tex]9.5\cos 36t^o=9.5-h[/tex][tex]h=9.5-9.5\cos 36t^o[/tex]The graph of the function is
A tourist from Mexico is vacationing in the U.S. One day he notices that gas cost 2.82dollars per gallon. On that same day, 1 peso is worth 0.053 dollars. Fill in the two blanks on the left side of equation using two of the ratios. Then write answer rounded to the hundredth on the right side of the equation
Answers
We must do the dimensional analysis to solve that problem, then, to have L in the denominator let's multiply $/gal by gal/L, if we do that
[tex]\frac{\$}{\text{ gal}}\cdot\frac{\text{ gal}}{\text{ L}}=\frac{\$}{\text{ L}}[/tex]Now we have a dollar per liter ($/L) we must change the dollar to pesos, then let's use pesos/$, it will result in
[tex]\frac{\$}{L}\cdot\frac{\text{ pesos}}{\$}=\frac{\text{ pesos}}{L}[/tex]Now we know the order we can just multiply the vales:, see that
[tex]\frac{\$}{\text{ gal}}\cdot\frac{\text{ gal}}{\text{ L}}\cdot\frac{\text{ pesos}}{\$}=\frac{\text{ pesos}}{L}[/tex]Then let's do it using the values
[tex]\frac{2.82\operatorname{\$}}{\text{gal}}\cdot\frac{\text{gal}}{3.79\text{L}}\cdot\frac{\text{pesos}}{0.053\operatorname{\$}}=14.04\frac{\text{pesos}}{L}[/tex]The final answer is
[tex]14.04\cdot\frac{\text{ pesos}}{L}[/tex]Let f(x)=-x-1 and g(x)=x^2-5 Find (f o g)(-2).
Answers
A composite function (f o g) (x) is the f[g(x)]
To find (f o g)(-2)
1) Let's find (f o g) (x) by substituing x by x² - 5 in f(x):
[tex]\begin{gathered} f(x)=x-1 \\ (f\circ g)(x)=(x^2-5)-1 \\ (f\circ g)(x)=x^2-5-1 \\ (f\circ g)(x)=x^2-6 \end{gathered}[/tex]2) To find (f o g) (-2), let's substitute x by 2 and solve the equation.
[tex]\begin{gathered} (f\circ g)(x)=x^2-6 \\ (f\circ g)(-2)=(-2)^2-6 \\ (f\circ g)(-2)=4^{}-6 \\ (f\circ g)(-2)=-2 \end{gathered}[/tex]Answer: -2.
Tell weather the ordered pair is a solution to the system of equations. you must show work to support your answer.
Answers
We are given the following system of equations
[tex]\begin{gathered} 2x-3y=4\quad eq.1 \\ 2x+8y=11\quad eq.2 \end{gathered}[/tex]We are asked to find out if the ordered pair (5, 2) is a solution to the given system of equations.
Let us substitute the given ordered pair into both equation and check if it satisfies both equations.
Substitute x = 5 and y = 2 into eq.1
[tex]\begin{gathered} 2x-3y=4\quad eq.1 \\ 2(5)-3(2)=4 \\ 10-6=4 \\ 4=4\quad (\text{satisfied)} \end{gathered}[/tex]Now, substitute x = 5 and y = 2 into eq.2
[tex]\begin{gathered} 2x+8y=11\quad eq.2 \\ 2(5)+8(2)=11 \\ 10+16=11 \\ 26\ne11\quad (\text{not satisfied)} \end{gathered}[/tex]As you can see, the given ordered pair do not satisify both of the equations therefore, it is not a valid solution to the given system of equations.
Select all the true statements. 2.5 pounds of peanuts costs $1. 1 pound of peanuts costs $2.50. 5 pounds of peanuts cost $12.50 9 pounds of peanuts cost $19.50. The point (4.10) is on the graph of the proportional relationship,
Answers
step 1
Find the equation of the proportional line
C=kW
where k is the constant of proportionality
we have the point (7,17.5)
k=C/W
k=17.5/7
k=2.5
so
the equation is
C=2.5W
Verify the statements
A) 2.5 pounds of peanuts costs $1
For W=2.5
Find the value of C
C=2.5(2.5)=$6.25
the statement is false
B) 1 pound of peanuts costs $2.50
For W=1
C=2.5(1)=$2.5
the statement is trueC) 5 pounds of peanuts cost $12.50
For W=5
C=2.5(5)=$12.5
the statement is trueD) 9 pounds of peanuts cost $19.50
For W=9
C=2.5(9)=$22.5
the statement is false
The line plot shows that you packed 10 boxes of different weights
Answers
The dot plot shows that 10 boxes were packed with different weights of books.
You have to determine how to repack the boxes so that each one has the same weight.
The first step is to determine the total weight of books you have to pack.
Looking at the dot plot, each x-mark indicates one box so that:
1 box has a weight of 1/8 pounds
3 boxes have a weight of 3/8 pounds
4 boxes have a weight of 4/8 pounds
2 boxes have a weight of 7/8 pounds
To calculate the total weight, you have to multiply each weight by the number of boxes that measure it and add all results, i.e. multiply each obervation (weight) by the observed frequency (number of boxes):
[tex]\begin{gathered} \text{Total weight=observed wight }\cdot\text{ nºboxes} \\ \text{Total weight=(1}\cdot\frac{1}{8}\text{)+(3}\cdot\frac{3}{8}\text{)+(4}\cdot\frac{4}{8}\text{)+(2}\cdot\frac{7}{8}\text{)} \\ \text{Total weight=5} \end{gathered}[/tex]All the books that were packed inside the 10 boxes add up to a total weight of 5 pounds.
Now that we know the total weight, to distribute it evenly in the 10 boxes you have to divide the total weight by the number of boxes:
[tex]\frac{\text{total weight}}{nºboxes}=\frac{5}{10}=\frac{1}{2}[/tex]To repack the boxes evenly, you have to pack 1/2 pound of books in each box. (2nd option)
Determine which operations polynomials, integers, and rationals are not closed under.a. Polynomials:b. Integers:c. Rationals:
Answers
We say that a set A is closed under a certain operation if
[tex]\begin{gathered} f,g\in A \\ \Rightarrow f\oplus g\in A \end{gathered}[/tex]In our case,
a) It is evident that the polynomials are closed under addition and subtraction
[tex]\begin{gathered} (a_mx^m+a_{m-1}x^{m-1}+\ldots+a_1x+a_0)+(b_nx^n+b_{n-1}x^{n-1}+\ldots+b_1x+b_0)_{} \\ =a_mx^m+\cdots(a_n+b_n)x_n+\cdots(a_0+b_0) \end{gathered}[/tex]Which is a polynomial. Similarly, in the case of the multiplication of polynomials.
However, in the case of the division of polynomials, the result is not always a polynomial.
[tex]\begin{gathered} \frac{x^2+3x+2}{x-1}\to\text{not a polynomial} \\ \frac{x^2+3x+2}{x+1}=x+3\to polynomial \end{gathered}[/tex]Thus, the set of polynomials is not closed under division.
b) Notice that the integers are closed under addition and subtraction (the integers are ...-2,-1,0,1,2,...).
[tex]\begin{gathered} a,b\in\text{integers} \\ \Rightarrow a+b\to\text{integer} \\ \end{gathered}[/tex]Similarly, in the case of the multiplication of integers.
Nevertheless, in the case of the division of integers,
[tex]\begin{gathered} \frac{6}{3}=2\to\text{integer} \\ \frac{7}{2}=3.5\to\text{not an integer} \end{gathered}[/tex]Therefore, the set of integers is not closed under division.
c) Rational numbers are numbers of the form a/b, where a and b are integers.
[tex]\begin{gathered} \frac{a}{b}+\frac{c}{d}=\frac{ad+bc\to\text{integer}}{bd\to integer}\to\text{rational number} \\ \frac{a}{b}\cdot\frac{c}{d}=\frac{ac\to integer}{bd\to\text{integer}}\to\text{rational number} \end{gathered}[/tex]Once again, the set of rational numbers is closed under addition, subtraction and multiplication. As for the division of rational numbers,
[tex]\frac{\frac{a}{b}}{\frac{c}{d}}=\frac{ad\to integer}{bc\to integer}\to rational\text{ number}[/tex]The rational numbers are also closed under division.
There are no restrictions for the set of rational numbers regarding operations.