Use geometry to evaluate (see attached) (calculus) the integral from 0 to 6 of the function f of x, dx for f of x equals 3 for x less than or equal to 3 and equals the quantity 6 minus x for x greater than 3.
27
13.5
12
10.5
Answers
notice the area is just a 3x3 square and a triangle with a base of 3 and a height of 3, simply get the area of both, add them up, and that's the area of the piece-wise on that range.
Answer:
13.5
Step-by-step explanation:
The geometric representation of an integral is the area below the curve. So, let's graphic the function.
[tex]f(x) = 3[/tex] is an horizontal line, (y = 3)
and [tex]f(x) = 6-x[/tex] is also a line, but it is decreasing.
We can see in the image that the area below the curve happens to be a polygon. In order to calculate the area we are going to separate it into simpler polygons: a square and a triangle.
The area of the square is [tex]3*3 = 9[/tex].
The area of the triangle is [tex]\frac{3*3}{2} = \frac{9}{2} = 4.5[/tex]
If we add them, we get
[tex]\int_0^6 f(x) dx =9 + 4.5 = 13.5[/tex]
Related Questions
Identify the properties needed to prove n + n + n + 2n = 5n.
Answers
The properties needed to prove n + n + n + 2n = 5n are: Distributive Property of Multiplication, Associative Property of Addition, and Commutative Property of Addition.
What is Commutative Property of Addition?The commutative property of addition states that when two or more numbers are added, the order in which they are added does not affect the sum. In other words, for any two numbers a and b, a + b = b + a.
The properties required to prove n + n + n + 2n = 5n are as follows:
Distributive Property of Multiplication: This property states that a(b + c) = ab + ac. In this case, we can apply the distributive property to 2n, which gives us 2n = n + n.Associative Property of Addition: This property states that (a + b) + c = a + (b + c). We can use the associative property of addition to group the n's together and the 2n's together, which gives us (n + n + n) + (2n) = 5n.Commutative Property of Addition: This property states that a + b = b + a. We can use the commutative property of addition to rearrange the terms in the equation, which gives us (n + n + n + 2n) = (n + n + 2n + n) = 5n.Thus, these properties can be used.
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Final answer:
To prove n + n + n + 2n = 5n, we use the commutative and associative properties of addition to group and combine like terms, then apply the distributive property to show that the sum equals 5n.
Explanation:
To prove the equation n + n + n + 2n = 5n, we need to use properties of equality and algebraic manipulation. First, we apply the commutative property of addition, which allows us to rearrange the terms in any order without changing the sum.
This property isn't strictly necessary here, as all terms involve 'n', but it's a useful property in general for manipulating equations.
Next, we utilize the associative property of addition, which allows us to group numbers in any combination when adding. In this specific case, we group all the 'n' terms together:
(n + n + n) + 2n
Using the associative property, we can simplify the equation further by combining like terms:
3n + 2n = 5n
Lastly, we apply the distributive property, which in this case shows that multiplying the coefficient (being 1 for n in this example) by the number of times 'n' appears (3 for the first set and 2 for the second set), gives the same result as adding 'n' that many times.
Find the point where the line crosses the x-axis. y = –5x + 5
Answers
Let y = 0, and solve for x.
y = -5x + 5
0 = -5x + 5
5x = 5
x = 1
The line crosses the x-axis at x = 1.
Write the expression 36+4 as a product using the greatest common factor and the distributive property.
Answers
(4 * 9) + (4 * 1) =
4(9 + 1) <==
The expression 36 + 4 can be written as a product using the greatest common factor and the distributive law is 4(9 + 1).
What are basic math properties?There are three basic math properties they are commutative property which states a + b = b + a, associative property which states
a + (b + c) = (a + b) + c and distributive property which states a(b+c) = ab + ac.
We know distributive property states a(b + c) = ab + ac.
Given 36 + 4, the here greatest common factor of 36 and 4 is 4.
∴ The expression 36 + 4 can be written as,
4(9 + 1).
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HELP ME WITH MATH PLEASEEEEEEEEEEEEE
Answers
(1^2x)/(2^2x) = 32
1^2x = 1
1/2^2x = 32
32 = 2^5
2^-2x = 2^5
take out the base, (since they are the same. You are left with...)
-2x = 5
-2x = 5 is your answer in this case
hope this helps :D
How many ways are there to distribute n distinct balls into r boxes (a box can contain any number of balls, ranging from zero (none) all the way to all n balls)?
Answers
There are 64 pretzels in a 16 ounce bag of chocolate covered pretzels. how many ounces does each pretzel way?
Answers
Simply Divide:
16/64 = 0.25
So, Each Pretzel Weighs 0.25 Ounces.
1.The state fair charges and entry fee plus $0.75 per ticket needed for the rides. Sally spent her money only on ride tickets and fair admission. The price of the fair admission is the same for everyone. Use y to represent the total cost and x to represent the number of ride tickets.
(a)Sally spent a total of $31.25 on the entry fee plus 25 ride tickets. How much did Sally pay for the entry fee?
Answer:
(b)Joe has $26 he can spend at the fair on the entry fee and tickets. How many tickets can Joe buy?
Answer:
(c)Let x represent the number of ride tickets purchased and y represent the total cost of fair entry fee and ride tickets. Write an equation to calculate the total cost of the entry fee and fair admission for a person who purchases x tickets.
Answer:
Answers
Final answer:
Sally paid a $12.50 entry fee to the fair. Joe can buy 18 tickets after paying the entry fee with his $26. The equation to represent the total cost of entry fee and tickets based on the number of tickets purchased is y = 12.50 + 0.75x.
Explanation:
(a) We are told Sally spent a total of $31.25 on the entry fee plus 25 ride tickets. Since each ticket costs $0.75, we can multiply 25 tickets by $0.75 to find the total cost of the tickets alone. That is 25 tickets × $0.75 per ticket = $18.75. Subtracting this amount from the total cost of $31.25 gives us the entry fee. The entry fee is $31.25 - $18.75 = $12.50.
(b) Joe has $26 to spend on the entry fee and tickets. We found out earlier that the entry fee is $12.50. Subtracting the entry fee from the total amount Joe has, we get $26 - $12.50 = $13.50. Now, divide this by the cost of one ticket, $0.75, to determine the number of tickets he can buy: $13.50 / $0.75 = 18 tickets.
(c) Let x represent the number of ride tickets purchased and y represent the total cost of fair entry fee and ride tickets. The total cost is the sum of the fixed entry fee and the variable cost of x tickets at $0.75 each. The equation is y = 12.50 + 0.75x.
What is the mean of the set of numbers?
-11, 22, -2, -3
Answers
add all the numbers and then divide by how many there are in the set to find the mean so
-11+22+-2+-3=6
6÷4=1.5
so 1.5 would be the mean
Hope this helps!
Hope this helped!
A quadratic factor that cannot be factored further as a product of linear factors containing real numbers is said to be
Answers
____________________________________
Write the equation of a line given two points kuta
Answers
Kate has 6 more dimes than she has nickels. She has a total of $3.60. How many coins does she have in all?
Answers
Suzanne has a student loan balance of $25,904 with an average interest rate of 6.8 percent. Use the student loan repayment calculator (Links to an external site.)Links to an external site. to determine how much her monthly payment be if she selects the standard repayment option.
Answers
Since the problem is requiring us to use the loan repayment calculator and here is what the calculator gave:
Loan Balance: $25,506.00
Adjusted Loan Balance: $25,506.00
Loan Interest Rate: 6.80%
Loan Fees: 0.00%
Loan Term: 10 years
Minimum Payment: $0.00
Monthly Loan Payment: $293.52
Number of Payments: 120 months
Cumulative Payments: $35,223.07
Total Interest Paid: $9,717.07
It is projected that you will need an annual salary of a minimum $35,222.40 to be capable to have enough money to repay this loan. This approximation assumes that 10% of your gross monthly income will be keen to repaying your student loans. This resembles to a debt-to-income ratio of 0.7. If you use 15% of your gross monthly income to repay the loan, you will need an annual salary of only $23,481.60, but you may experience some financial difficulty. This corresponds to a debt-to-income ratio of 1.1.
Using the order of precedence for arithmetic operators, these two formulas will calculate the same result: =22/2*5 =22/(2*5).
Answers
22/2 x 5 = 11 x 5 = 55
22/(2 x 5) = 22 / 10 = 2.2
So, you see both have different answers.
Which of the following demonstrates the commutative property across multiplication for 2 × 10 × 7 A. 2 + 10 × 7 B. 2 × (10 × 7) C. 10 ×7 × 2 D. (2 × 10) × 7
Answers
The correct answer is C. 10×7×2
Answer:
Option C is correct
[tex]10 \times 7 \times 2[/tex]
Step-by-step explanation:
Commutative property of multiplication states that you can multiply number in any order.
For example:
[tex]2 \times 3 = 3 \times 2[/tex]
[tex]3 \times 4 \times 5 = 4 \times 5 \times 2[/tex]
Given that:
2 × 10 × 7
By definition Commutative property of multiplication,
⇒[tex]10 \times 7 \times 2[/tex]
Therefore, [tex]10 \times 7 \times 2[/tex] demonstrates the commutative property across multiplication for 2 × 10 × 7
The area of a square boxing ring is 436 square feet. What is the approximate length, in feet, of one side of the boxing ring?
4ft
21ft
54ft
109ft
Answers
The area of the square is the square of the side thus the approximate length of the square will be 21 ft so option (B) will be correct.
What is a square?A square is a geometrical figure in which we have four sides each side must be equal and the angle between two adjacent sides must be 90 degrees.
Area of square = Side²
As per the given question,
The area of a square boxing ring is 436 square feet.
It is known that,
Area = Side²
Side² = 436
Take square root on both sides,
Side = √436 = 20.88 ≈ 21 ft
Hence "The area of the square is the square of the side thus the approximate length of the square will be 21 ft so option (B) will be correct".
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Jamal is driving in Europe and finds that he has to drive 56 kilometers until he arrives at his destination. If 1 km = 0.62137 mi, about how many miles does Jamal have left on his journey?
Answers
Jamal has to drive 56 kilometers until he arrives at his destination, so he needs to drive 56*0.62137 = 34.7967 mi.
So Jamal have 34.7967 miles left.
Hope this helps! :)
What is the value of x? Enter your answer in the box.
Answers
The value of x from the equilateral triangle is x = 12 and the side is 38
What is an Equilateral Triangle?
An equilateral triangle is a triangle in which all three sides have the same length.
Let the triangle be ΔABC , and
∠A = ∠B = ∠C = 60° and AB = BC = CA
Given data ,
Let the triangle be represented as ABC
Now , the triangle ABC is an equilateral triangle with
∠A = ∠B = ∠C = 60°
The measure of side AB = 4x - 10
The measure of side BC = 3x + 2
The measure of side AC = 5x - 22
And , the sides of the triangle will also be the same
So ,
Side AB = Side BC = Side CA
Substitute the values in the equation , we get
4x - 10 = 3x + 2 be equation (1)
On simplifying the equation , we get
Adding 10 on both sides of the equation , we get
4x = 3x + 12
Subtracting 3x on both sides of the equation , we get
x = 12
And ,
5x - 22 = 3x + 2
Adding 22 on both sides of the equation , we get
5x = 3x + 24
Subtracting 3x on both sides of the equation , we get
2x = 24
Divide by 2 on both sides of the equation , we get
x = 12
Hence , the value of x is 12
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Drive-in movies were popular in the 1930s and 1940s. In 1933, there were 100 drive-in theater in the U.S by 1945 there were 2200. What was the percent of increase from 1933 to 1945?
Answers
To find the percent of increase in the number of drive-in theaters from 1933 to 1945, subtract the initial number from the final number, divide by the initial number, and multiply by 100, resulting in a 2100% increase.
Explanation:The question asks how to calculate the percent of increase in the number of drive-in theaters in the U.S. from 1933 to 1945. To find the percent of increase you subtract the original number from the new number divide by the original number, and then multiply by 100 to convert to a percentage.
Original number (1933) = 100 drive-in theatersNew number (1945) = 2200 drive-in theatersPercent of increase = ((2200 - 100) ÷ 100) × 100Percent of increase = ((2100) ÷ 100) × 100 = 2100%Therefore, the percent of increase from 1933 to 1945 was 2100%.
The perimeter of a scalene triangle is 14.5 cm. The longest side is twice that of the shortest side. Which equation can be used to find the side lengths if the longest side measures 6.2 cm?
Answers
Perimeter is the sum of all sides of a polygon. For this case, the perimeter is expressed as: P=a+b+c. It has been said that the longest side is twice that of the shortest side. Let's assume that a is the longest side and b is the shortest side. So, the equation for the shortest side is,
6.2 = 2c
c = 3.1
14.5 = 6.2 + 3.1 + b
14.5 = 9.3 + b
I take no credit for this answer. It was solved by Taskmasters.
Which function represents the graph of f(x)=|2x| after it is translated 5 units to the left?
Answers
Answer:
Here are the options to the questions:
a.) g(x)=|2(x−5)|
b.) g(x)=|2(x+5)|
c.) g(x)=|2x+5|
d.) g(x)=|2x−5|
The answer is b.) g(x)=|2(x+5)|
Step-by-step explanation:
The function F(x) = |2x| has the vertex at point (0,0)
The rule of translation is (x,y) -------- (x - 5, y)
The translation of the point (0,0)is equal to
(x,y) -------- (0 - 5,0)
(x,y) -------- (-5,0)
The function g(x) has the vertex at point (-5,0)
So therefore, the function g(x) is equal to g(x)=|2(x+5)|
PLS HELP!!! WILL MAKE BRAINLIEST!!!
Answers
First, start by distributing the six by each fraction. (Multiply 6 by each number inside the parenthesis)
6*1/2= 3
6*2/3= 4
6*3/4= 4 1/2
Then add all of the solutions together.
3+4+4.5= 11 1/2
Calculation without Distribution
Add all the fractions together then multiply by 6.
1/2+2/3+3/4= 1 11/12
1 11/12* 6= 11 1/2
Hopefully, I could help!
−3x^ 4−7x+17=0
2
3
4
5
How many complex roots does the polynomial equation have?
(Show work if possible)
Answers
There are two complex roots of the polynomial equation.
What is a polynomial?They are mathematical expressions involving variables raised with nonnegative integers and coefficients and constants with only operations of addition, subtraction, multiplication, and nonnegative exponentiation of variables involved.
We have been given a polynomial expression as;
[tex]-3x^ 4-7x+17=0[/tex]
Since the Degree of a polynomial is the highest power that its terms (for multi-variables, the power of the term is the addition of the power of variables in that term).
Therefore, the highest power of the given polynomial is 4.
Thus, the Maximum possible roots are 4 but; only 2 are real and 2 are complex.
Hence, There are two complex roots of the polynomial equation.
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Find the 5th term of a sequence represented by f(n) = 5n-2
Answers
5n - 5
5(5) - 5 25 - 5
f(n) = 5
Which value is the best estimate of 26% of 259?
A: 50
B: 52
C: 65
D: 88
Answers
PROOF
just multiply
259×0.26=67.34
What is the square root of sixty four
Answers
Lillian wants to buy a digital camera for $143.00. She baby-sits for $2.75 an hour. So far she has worked a total of 30 hours. How many more hours does she need to work to have enough to buy the camera?
Answers
Put the numbers in order from least to greatest.
17
, 4.12, 41
10
, 42
15
A) 17
, 4.12, 41
10
, 42
15
B) 41
10
, 17
, 4.12, 42
15
C) 41
10
, 4.12, 17
, 42
15
D) 42
15
, 4.12, 17
, 41
Answers
Simplify m - {n - [p + (m + n - p)]}.
0
2m
-2m + 2p
Answers
= m - {n - [p + m + n - p]}
= m - {n - m - n}
= m + m
= 2m
answer
2m
Answer:To simplify the following expression, we first simplify the terms inside the parenthesis from inner going outer. The first one is [p - (m + n - p)] equal to p - m- n + p = 2p - m - n. Then, n +2p - m - n equal to 2p -m. Then, m - 2p + m equal to 2m -2p. Answer is B
Step-by-step explanation:
What is equal to 3log2 + log4
Answers
Log Rule #2: log(x^y) = y*log(x)
Using those log rules, we can say...
3*log(2) + log(4) = log(2^3) + log(4) <-- use log rule #2
3*log(2) + log(4) = log(8) + log(4) <-- simplify 2^3 to get 8
3*log(2) + log(4) = log(8*4) <-- use log rule #1
3*log(2) + log(4) = log(32) <-- simplify 8*4 to get 32
So the expression simplifies to log(32)
Given the function f(x) = 2(x + 6), find x if f(x) = 22. PLEASE ANSWER DONT PLAY I REPORT NO GAME PLAYING PLEASE
Answers
x + 6 = 11
x = 5
check
f(5) = 2(5 + 6) = 2(11) = 22
Answer:
The correct answer is 5
Step-by-step explanation: