Answer:
y-determinant = 2
Step-by-step explanation:
Given the following system of equation:
x + 2y = 5 x - 3y = 7Let's represent it using a matrix:
[tex]\left[\begin{array}{ccc}1&2\\1&-3\end{array}\right] = \left[\begin{array}{ccc}5\\7\end{array}\right][/tex]
The y‐numerator determinant is formed by taking the constant terms from the system and placing them in the y‐coefficient positions and retaining the x‐coefficients. Then:
[tex]\left[\begin{array}{ccc}1&5\\1&7\end{array}\right] [/tex]
y-determinant = (1)(7) - (5)(1) = 2.
Therefore, the y-determinant = 2
Logarithms- How to answer these questions?
Answer:
2. m = b³ (= 216)
3. logp(x) = -4
Step-by-step explanation:
2. The given equation can be written using the change of base formula as ...
... log(m)/log(b) + 9·log(b)/log(m) = 6
If we define x = log(m)/log(b), then this becomes ...
... x + 9/x = 6
Subtracting 6 and multiplying by x gives ...
... x² -6x +9 = 0
... (x -3)² = 0 . . . . . factored
... x = 3 . . . . . . . . . value of x that makes it true
Remembering that x = log(m)/log(b), this means
... 3 = log(m)/log(b)
... 3·log(b) = log(m) . . . . . multiply by the denominator; next, take the antilog
... m = b³ . . . . . . the expression you're looking for
___
3. Substituting the given expression for y, the equation becomes ...
... logp(x^2·(p^5)^3) = 7
... logp(x^2) + logp(p^15) = 7 . . . . . use the rule for log of a product
... 2logp(x) + 15 = 7 . . . . . . . . . . . . . use the definition of a logarithm
... 2logp(x) = -8 . . . . . . . . . . . . . . . . subtract 15
... logp(x) = -4 . . . . . . divide by 2
Answer:
For #3: [tex]\log_px=-4[/tex]
Step-by-step explanation:
I'm a little rusty on my logarithm rules for #2, but here's an explanation of #3.
Logarithms: the Inverse of ExponentsIn a sense, we can think of operations like subtraction and division as different ways of representing addition and multiplication. For instance, the same relationship described by the equation 2 + 3 = 5 is captured in the equation 5 - 3 = 2, and 5 × 2 = 10 can be restated as 10 ÷ 2 = 5 without any loss of meaning.
Logarithms do the same thing for exponents: the expression [tex]2^3=8[/tex] can be expressed in logarithms as [tex]\log_28=3[/tex]. Put another way, logarithms are a sort of way of pulling an exponent out onto its own side of the equals sign.
The ProblemOur problem gives us two facts to start: that [tex]log_p(x^2y^3)=7[/tex] and [tex]p^5=y[/tex]. With that, we're expected to find the value of [tex]\log_px[/tex]. [tex]p^5=y[/tex] stands out as the odd-equation-out here; it's the only one not in terms of logarithms. We can fix that by rewriting it as the equivalent statement [tex]log_py=5[/tex]. Now, let's unpack that first logarithm.
Justifying Some Logarithm RulesFor a refresher, let's talk about some of the rules logarithms follow and why they follow them:
Product Rule: [tex]\log_b(MN)=\log_bM+\log_bN[/tex]
The product rule turns multiplication in the argument (parentheses) of a logarithm into addition. For a proof of this, consider two numbers [tex]M=b^x[/tex] and [tex]N=b^y[/tex]. We could rewrite these two equations with logarithms as [tex]\log_bM=x[/tex] and [tex]\log_bN=y[/tex]. With those in mind, we could say the following:
[tex]\log_b(MN)=\log_b(b^xb^y)[/tex] (Substitution)[tex]\log_b(b^xb^y)=log_b(b^{x+y})[/tex] (Laws of exponents)[tex]\log_b(b^{x+y})=x+y[/tex] ([tex]\log_b(b^n)=n[/tex])[tex]x+y=\log_bM+\log_bN[/tex] (Substitution)And we have our proof.
Exponent Rule: [tex]\log_b(M^n)=n\log_bM[/tex]
Since exponents can be thought of as abbreviations for repeated multiplication, we can rewrite [tex]\log_b(M^n)[/tex] as [tex]\log_b(M\times M\cdots \times M)[/tex], where M is being multiplied by itself n times. From there, we can use the product rule to rewrite our logarithm as the sum [tex]\log_bM+\log_bM+\cdots+\log_bM[/tex], and since we have the term [tex]\log_bM[/tex] added n times, we can rewrite is as [tex]n\log_bM[/tex], proving the rule.
Solving the ProblemWith those rules in hand, we're ready to solve the problem. Looking at the equation [tex]\log_p(x^2y^3)=7[/tex], we can use the product rule to split the logarithm into the sum [tex]\log_p(x^2)+\log_p(y^3)=7[/tex], and then use the product rule to turn the exponents in each logarithm's argument into coefficients, giving the equation [tex]2\log_px+3\log_py=7[/tex].
Remember how earlier we rewrote [tex]p^5=y[/tex] as [tex]log_py=5[/tex]? We can now use that fact to substitute 5 in for [tex]log_py[/tex], giving us
[tex]2\log_px+3(5)=7[/tex]
From here, we can simply solve the equation for [tex]\log_px[/tex]:
[tex]2\log_px+15=7\\2\log_px=-8\\\\\log_px=-4[/tex]
Drag each tile to the correct box.
A scientist is studying the growth rates of three samples of bacteria in different conditions. The following three functions represent the number of bacteria in the three samples after x hours.
Order the samples by their average growth rate over the interval [1, 3], from least to greatest.
Answer: C < B < A
Step-by-step explanation:
First, let's evaluate the growth rate of A from [1, 3]
[tex]f(x) = 200\bigg(\dfrac{3}{2}\bigg)^1[/tex]
= 300
coordinate is (1, 300)
[tex]f(x) = 200\bigg(\dfrac{3}{2}\bigg)^3[/tex]
= 675
coordinate is (3, 675)
Average growth rate (AGR) is: [tex]\dfrac{675-300}{3-1} = \dfrac{375}{2} = 187.5[/tex]
AGR (A) = 187.5
*************************************************************************************
Next, let's evaluate the growth rate of B from [1, 3]
The coordinates are already provided as (1, 120) and (3, 480)
Average growth rate (AGR) is: [tex]\dfrac{480-120}{3-1} = \dfrac{360}{2} = 180[/tex]
AGR (B) = 180
*************************************************************************************
Lastly, let's evaluate the growth rate of C from [1, 3]
f(x) = 600(1.2)ˣ
f(x) = 600(1.2)¹
= 720
coordinate is (1, 720)
f(x) = 600(1.2)³
= 1036.8
coordinate is (3, 1036.8)
Average growth rate (AGR) is: [tex]\dfrac{1036.8-720}{3-1} = \dfrac{316.8}{2} = 158.4[/tex]
AGR (C) = 158.4
Brian rented a book from the book shop.The shop charges $2 for the first day and $3 from the second day onwards.Brian took more then 2 days to return the book and he paid $35 while returning the book.How many days did he keep the book with him?
Brian rented the book for 12 days.
Brian rented a book with the condition that the shop charges $2 for the first day and $3 from the second day onwards. He paid a total of $35 upon returning the book. To determine how many days he kept the book, we must calculate the cost for each day and subtract the initial $2 from the total amount paid.
Steps to Calculate the Number of Days:
Subtract the first day's charge from the total amount: $35 - $2 = $33.Divide the remaining amount by the daily charge starting from the second day: $33 / $3 = 11 days.Add the first day to the result: 11 days + 1 day = 12 days.Therefore, Brian kept the book with him for 12 days.
Lucy cuts 4 squares with side length x in. from the corners of a 12 in. by 18 in. cardboard rectangle. She folds the remaining cardboard to make a tray that is x in. high. Write and simplify a polynomial function for the volume V of the tray in terms of x.
please help, i am loosing a braincell each day
Answer:
Volume of tray =[tex]4x^{3} -60x^{2} +216x[/tex]
Step-by-step explanation:
The Dimensions of rectangular cardboard is given by 12 inches by 18 inches
Length of the rectangular cardboard = 18 inches
Width of the rectangular cardboard = 12 inches
if the square of sides x inches is cut from each corner of the rectangular cardboard and folded to make a tray, then we have
Length of the tray = length of cardboard - 2 (side of the square )
= [tex]18-2x[/tex]
Width of the tray = width of cardboard - 2( side of square)
= [tex]12-2x[/tex]
height of tray = sides of square
= x
volume of tray = Length × width × height
Volume of tray = [tex]x(18-2x)(12-2x)[/tex]
first we multiply (12-2x) and (18-2x)
Volume of tray =[tex]x(18(12-2x)-2x(12-2x))\\[/tex]
=[tex]x(216-36x-24x+4x^{2})[/tex]
=[tex]x(4x^{2}-60x+ 216)[/tex]
=[tex]4x^{3} -60x^{2} +216x[/tex]
Hence the volume of tray =[tex]4x^{3} -60x^{2} +216x[/tex]
The volume of the Tray is given by the equation [tex]4x^3-60x^2+216x[/tex] and this can be determined by using the given data.
Given :
Lucy cuts 4 squares with side length x in. from the corners of a 12 in. by 18 in. cardboard rectangle. She folds the remaining cardboard to make a tray that is x in.If the square is cut from each corner of the cardboard and folded to make a tray, then according to the given data:
Tray length = 18 - 2x
Tray width = 12 - 2x
Tray height = x
Now, the volume of the tray is given by the expression:
The volume of Tray = [tex]x(12-2x)(18-2x)[/tex]
Simplify, the above expression.
[tex]\rm Volume = (12x - 2x^2)(18-2x)[/tex]
[tex]\rm Volume = 216x -24x^2-36x^2+4x^3[/tex]
[tex]\rm Volume = 4x^3-60x^2+216x[/tex]
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what is 3/20 as a percentage
Answer:
15%
Step-by-step explanation:
Percent means out of 100. Lets get the denominator out of 100
3/20 * 5/5 = 15/100
This is 15%
Final answer:
To convert the fraction 3/20 to a percentage, first convert it to a decimal by dividing 3 by 20. Then, multiply the decimal by 100 to get the percentage.
Explanation:
To express 3/20 as a percentage, you can multiply the fraction by 100. The calculation is (3/20) * 100 = 15%. Therefore, 3/20 as a percentage is 15%. This means that 3 out of every 20 parts represent 15% of the whole. In percentage terms, it provides a convenient way to compare the fraction to 100, making it easier to understand its relative size or portion in relation to the total. In this case, 15% indicates the proportion of the whole that is represented by the fraction 3/20.
Laura borrows 10000 for 5 years at 6.5% rate. How much will the interest be after 3 years?
SUPER EASY AND REALLY URGENT!!
Answer:
is your answer $3250
Step-by-step explanation:
so we dont know our I (interest) but :
P (principal) = 10000
R (rate) = 6.5% we'll have to turn it into the decimal which is 0.065
T (time) = 5 years
and our basic formula of finding I is I = PxRxT so :
I = 10000 x 0.065 x 5
I = 3250
The letters in the word Mississippi are put into a bag. What’s the probability of randomly selecting a vowel from the bag ?
- 3/11
- 2/11
- 1/11
- 4/11
What multiplication sentence does the model represent?????
Answer:
I Think It Is 0.12*2
Because I See Both Of The Shadings Colored Differently And Each One Has 12 That Is The Reason For Me Thinking This.
Multiply. Express your answer in simplest form. 7/10 × 2/5
Answer:
7/10 x 2/5 = 14/50 = 7/25
Step-by-step explanation:
mutipily 7 and 2 that gives you 14 and mutilpily 10 and 5 that gives you 50 = 14/50. In simplest form the answer is 7/25.
Answer:
7/10 x 2/5 = 14/50 = 7/25
Step-by-step explanation:
mutipily 7 and 2 that gives you 14 and mutilpily 10 and 5 that gives you 50 = 14/50. In simplest form the answer is 7/25.
Which of the function below is that inverse of f(x)=x^-25
Answer:
g⁻¹(x) = ±√(x + 25)
Step-by-step explanation:
g(x) = x² - 25 Rename g(x) as y
y = x² - 25
=====
Solve for x
y = x² - 25 Add 25 to each side
y + 25 = x² Take the square root of each side
x = ±√(y + 25) Switch x and y
y = ±√(x + 25) Rename y as g inverse
g⁻¹(x) = ±√(x + 25)
=====
See the graphs of g(x) and g⁻¹(x) below.
g(x) is the red line. g⁻¹(x) is the purple line.
Each graph reflects the other about the dashed line representing the function y = x.
which expression is equivalent to 4 times the sum of 3 and x
Answer:
4(3+x)
Step-by-step explanation:
If any thing is in the brackets you need to do it first
so if you do 3 + x × 4 it will give yo 4x + 3 or 3 +4x
The given expression 4 (x + 3) is equivalent to the expression 4x + 12.
What is an equivalent function?The equivalent operations are those that have various forms but have the same outcome.
The expression is given below.
→ 4 (x + 3)
The expression can be written as
→ 4x + 12
The given expression 4 (x + 3) is equivalent to the expression 4x + 12.
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Leonard paid $35 registration fee to join the gym. He pays $10 each month.
How much will he pay to use the gym for 6 months?
A customers photo cell phone bill this month is $50 and 50 Cent the company charges a monthly fee of $18 +5 cents for each car used in to represent the number of calls
What is the unit price of 3 gallons of juice for 6.99
Answer:
The unit price is $2.33 per gallon
Step-by-step explanation:
To find the unit price, we take the price and divide by the amount
$6.99 / 3 gallons
$2.33 per gallon
The unit price is $2.33 per gallon
45 POINTS!
PLEASE HELPPPPPPPPPPPPPP!
Answer:
This is a direct variation where the constant of variation is 5/9
Step-by-step explanation:
The equation for direct variation is y = kx
5x-9y =0
Lets try to get this equation in the above form
Add 9y to each side
5x-9y +9y = 9y
5x = 9y
Divide each side by 9
5/9 x = 9y/9
5/9x = y
This is in the form y = k x where k =5/9
in the figure above CD is the perpendicular bisector of AB. three students explained how they proved ADC is congruent to BDC
who's explanation is incorrect??
Answer:
All explanations are correct
Step-by-step explanation:
In triangle ABC, CD is the perpendicular bisector of AB, thus using ΔADC and ΔBDC,
AC=BC ( Since CD is perpendicular to AB, therefore C is equidistant from both A and B)
∠ADC=∠BDC=90°(CD perpendicular AB)
CD=CD( Reflexive property)
therefore, by SAS rule of congruency,
ΔADC≅ ΔBDC,
Also, In the same triangles, AC=BC ( Since CD is perpendicular to AB, therefore C is equidistant from both A and B)
CD=CD( Reflexive property)
AD=BD (D is the midpoint and divides AB into two equal halves)
Thus, by SSS rule of congruency,
ΔADC≅ ΔBDC,
Thus, all the three explanations are correct.
In the given case, we can conclude that All explanations are correct
In triangle ABC, let CD be the perpendicular bisector of AB. We can use the properties of triangles to demonstrate that ΔADC is congruent to ΔBDC.
AC=BC: This holds true because CD is perpendicular to AB, making point C equidistant from both A and B.
∠ADC=∠BDC=90°: Since CD is perpendicular to AB, both angles ADC and BDC are right angles.
CD=CD: This is a reflexive property, stating that any line segment is equal to itself.
By applying the SAS (Side-Angle-Side) rule of congruency using the above properties, we can conclude that ΔADC is congruent to ΔBDC.
Furthermore, we can establish that in these congruent triangles:
AC=BC (since C is equidistant from A and B due to CD being perpendicular to AB).
CD=CD (reflexive property).
AD=BD (as D is the midpoint of AB).
Hence, using the SSS (Side-Side-Side) rule of congruency, we can also conclude that ΔADC is congruent to ΔBDC.
Therefore, all three explanations correctly demonstrate the congruence of these triangles.
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A motorcycle can travel 70 miles per gallon. Approximately how many gallons of fuel will the motorcycle need to travel 60 km?
[1 mile = 1.6 km]
a- 0.2
b- 0.5
c- 0.8
d- 1.4
Answer:
We have the following given
mileage = 70 miles per gallon
distance = 60 km
Required: amount of fuel in gallons
First, we must convert the distance to miles, so
60 km (1 mi/1.6 km) = 37.5 mi
So, the amount of fuel is
37.5 mi / (70 mi/gal) = 0.54 gallon
Answer:
B
Step-by-step explanation:
70 miles * [1.6 km/mile] = 112 km
Notice when you multiply by km/mile by miles the miles cancel out.
1 gallon will get 112 km
x gallon will get 60 km
1/x = 112/60 cross multiply1 * 60 = 112 * x divide by 11260/112 = x switchx = 60/112 Do the divisionx = 0.54 Which rounds to 0.5What is 2pi/ 5radians in degrees
Answer:
2/5 * pi radians = 72 degrees
Step-by-step explanation:
To convert radians to degrees, we multiply by 180/pi
2 * pi/5 * 180/pi =
The pi/pi cancels leaving
2/5 * 180
360/5
72
2/5 * pi radians = 72 degrees
what is the
the sum of the multiples of 35 rom 36 to 153
Answer:
70, 105, 140
Step-by-step explanation:
Park Crest Middle School has a population of about 400 8th graders. Parkcrest is one of several middle schools in the state which has a total of 806, 240 8th graders. Use powers of 10 to estimate about how many times greater the States population of eighth-graders is compared to the number of 8th graders at Park Crest Middle School.
Answer:
[tex]8.05 * 10^{5}[/tex]
Step-by-step explanation:
Thinking process:
the population of Park Crest Middle School = 400 children
The population in the state = 806, 240 children
The difference:
[tex]806240 - 400\\= 805840[/tex]
Expressing the difference in standard form gives:
8.05 × 10⁵
share $500.00 between four children in the ratio 2:3:5:10.
the cheap answer is, we divide 500 by (2+3+5+10) and then distribute accordingly.
[tex]\bf \cfrac{500}{2+3+5+10}\implies \cfrac{500}{20}\implies \cfrac{25}{1}\implies 25 \\\\\\ \stackrel{2\cdot 25}{2}~~:~~\stackrel{3\cdot 25}{3}~~:~~\stackrel{5\cdot 25}{5}~~:~~\stackrel{10\cdot 25}{10}\qquad \implies \qquad 50~~:~~75~~:~~125~~:~~250[/tex]
Which number sentence is true
Answer:
B is the right one i bet u 100 point s
Answer:
B is the correct answer.
The brackets ║ represent absolute value. Absolute value is ALWAYS positive. I hope this helps you!
-Mikayla
There are 225 dozen cookies in the bakery. How many cookies are there?
Answer:
2700 cookies.
Step-by-step explanation:
A dozen is a grouping of twelve, that is a dozen = 12.
If there are 225 dozen cookies in the bakery, then there are
[tex]225\cdot 12=2700[/tex] cookies in the bakery.
there are 2700 cookies in total
Since there are 225 dozen cookies in the bakery, we need to determine the total number of cookies.
To convert dozen to individual units, we multiply the number of dozens ( 225 ) by 12, as there are 12 cookies in each dozen :
225 dozen * 12 cookies / dozen = 2700 cookies
Therefore, there are 2700 cookies in total. By multiplying the number of dozens by 12, we account for the fact that each dozen contains 12 cookies. This calculation allows us to determine the precise quantity of cookies in the bakery.
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Mrs Wong is going on a trip. She has 14 books that she hasn’t read yet but she wants to bring only 2 on the trip. In how many ways can she choose 2 books to bring on the trip
In combinatorics, we use the combination formula to calculate the number of ways Mrs. Wong can choose 2 books out of 14. The result is 91 ways.
Explanation:The subject of this question is combinatorics, a topic in mathematics dealing with combinations of objects belonging to a finite set in accordance with certain constraints, such as those specified in this question. In this case, Mrs. Wong has 14 special objects (books) and wants to choose 2 out of them.
We use the combination formula in this scenario. The combination formula is given by C(n, r) = n! / [(n-r)!*r!], where n represents the total number of objects, r is the number of objects to choose, and '!' denotes factorial.
Substituting n as 14 and r as 2 into the formula, we get: C(14,2) = 14! / [(14-2)!*2!] = 91.
Therefore, Mrs. Wong can choose 2 books out of 14 in 91 different ways.
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The question is about combinations in mathematics. Using the formula for combinations, we find that Mrs. Wong can choose 2 books out of 14 in 91 different ways.
Explanation:This is a problem of combinations in mathematics. Mrs. Wong can choose two out of 14 unread books in a certain number of ways, and we're tasked to find that number.
Considering that order of selection does not matter, we can use the formula for combination: nCr = n! / r!(n-r)!. Here, 'n' is the total number of items, 'r' is the items to be chosen.
So the combinations she can make, denoted as 14C2, can be calculated like this:
14C2 = 14! / 2!(14-2)! = (14*13) / (2*1) = 91 ways
Therefore, Mrs Wong can choose 2 books out of 14 in 91 ways.
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what three things must a ordered pair have?
Answer: An x and y value, parenthesis around the values, and a comma in between them.
Examples of ordered pairs: (2, 5), (1872, 1683), (-3, 7), (7/8, -10/2)
Morita works in a florist shop and makes flower arrangements. She outs 10 flowers in each case, and 2/10 of the flowers are daisies. If Morita makes 4 arrangements, how many daisies does she need. Show how you can check your answer. Last weekend, Morita used 10 daisies to make flower arrangements. How many flowers other than daisies did she use to make the arrangements? Explain your reasoning.
Answer:
8 daisies
Step-by-step explanation:
You have to multiply the amount of vases by daisies.
Answer:
Givens
Morita outs 10 flowers in each case.2/10 of the flowers are daisies.Morita makes 4 arrangements.We know that Morita arranges 4 cases, each of them has 10 flowers.
So, if there are 2/10 daisies per case, that means each can contains 2 daisies.
But, she did 4 arrangements. There are 2x4 = 8 daisies in total.
Now, last weekend, Morita used 10 daisies to make flower arrangements, and we know she tend to use 2/10 of the flowers as daisies, that means this fraction represents daisies,
[tex]\frac{2}{10}x=10\\ x=\frac{100}{2}\\ x=50[/tex]
Therefore, there are 50 flowers in total.
I is the origin and P is the (4,3). Rx and Ray are reflections around the x- and y- axes
(4,-8)
(-4,8)
(-4,-8)
Answer:
The correct option is 1. The image of (2,4) is (4,-8).
Step-by-step explanation:
The given rule is
[tex]R_x{\circ}D_{o,2}:(2,4)[/tex]
The transformations perform from right to left. [tex]D_{o,2}[/tex] means dilation with scale factor 2 and center of dilation is origin.
The given rule defines the dilation with scale factor 2 and center of dilation is origin followed by reflection across x-axis.
If a figure dilated by scale factor k and the center of dilation is origin, then
[tex](x,y)\rightarrow (kx,ky)[/tex]
The scale factor is 2,
[tex](x,y)\rightarrow (2x,2y)[/tex]
[tex](2,4)\rightarrow (4,8)[/tex]
If a figure reflected across x-axis, then x-coordinate remains the same but the sign of y-coordinate is changed.
[tex](x,y)\rightarrow (x,-y)[/tex]
[tex](4,8)\rightarrow (4,-8)[/tex]
Therefore image of (2,4) is (4,-8) and option 1 is correct.
What is the 41st term in sequence 8,10,12,14,.....?
find the area of washers whose external and internal diameter are 1.2m and 0.9m
Answer:
0.49 m²
Step-by-step explanation:
The area (A) is calculated using A = πr² ( r is the radius )
A = external area - internal area
external radius = [tex]\frac{1.2}{2}[/tex] = 0.6
internal radius = [tex]\frac{0.9}{2}[/tex] = 0.45
A = π(0.6² - 0.45²)
= π(0.36 - 0.2025) = 0.1575π ≈ 0.49 m²
Final answer:
The area of the washers is 0.49m².
Explanation:
To find the area of the washers, we need to find the areas of the outer and inner circles and then subtract the smaller area from the larger one. The area of a circle is given by the formula A = πr², where r is the radius of the circle. In this case, the external diameter is 1.2m, so the radius is half of that, or 0.6m. The area of the outer circle is A₁ = π(0.6)² = 1.13m². Similarly, the radius of the inner circle is 0.45m, and the area of the inner circle is A₂ = π(0.45)² = 0.64m². Finally, the area of the washer is the difference between the two areas: A = A₁ - A₂ = 1.13m² - 0.64m² = 0.49m².
convert angle 180° to radians
Answer:
180 degrees in radians is π
Step-by-step explanation:
0° = 0 radians
90°= π/2
180 °= π
270°= 3π/2
360° = 2π
and so on