You meant pi, pi is a irrational number, but they use 3.14 for approximation, so that is your answer.
Hope this helped!
Nate
[tex]\pi-\bold{pi}\\\\\pi\ \text{it's the ratio of a circle's circumference to its diameter}\\\\\pi=\dfrac{\text{circumference of a circle}}{\text{circle diameter}}\\\\\pi\ \text{it's irrarional number}\\\\\pi=3.14159265358979323846264338327...\\\\\text{Approximation of}\ \pi:\\\\\pi\approx3.14\\\\\pi\approx\dfrac{22}{7}\\\\\pi\approx\dfrac{355}{113}[/tex]
The number pi occurs when calculating the surface area or volume of the rotational solids.
Cylinder:
[tex]S.A.=2\pi r^2+2\pi rH\\\\V=\pi r^2H[/tex]
Cone:
[tex]S.A.=\pi r^2+\pi rl\\\\V=\dfrac{1}{3}\pi r^2H[/tex]
Sphere:
[tex]S.A.=4\pi R^2\\\\V=\dfrac{4}{3}\pi R^3[/tex]
The number pi occurs when calculating the area and the circumference of a circle.
[tex]C=\pi d=2\pi r\\\\A=\pi r^2[/tex]
It is also used to convert angle measure in degrees to radians.
[tex]x^o=\dfrac{x\pi}{180}\ rad[/tex]
determine the divisibilty of the following 6501
Answer:
Here is a complete list of numbers that 6501 is divisible by:
1, 3, 11, 33, 197, 591, 2167, 6501
Step-by-step explanation:
I hope that was the answer you were looking for have a nice day :p
The number 6501 is not divisible by common divisors like 2, 3, 4, etc., and is in fact a prime number.
To determine the divisibility of the number 6501, we examine the number against known divisibility rules. However, unlike numbers like 4, 12, and 11, which have easy-to-apply divisibility rules, 6501 does not have an obvious rule that we can apply. So, we have to actually perform the division or use a calculator if we are trying to determine its divisibility by numbers other than 1 and itself.
For example:
The number 6501 is divisible by 1 and 6501 (since all numbers are divisible by themselves and 1).
To check for divisibility by other numbers, use division (for example: 6501 ÷ 2 is not an integer, so it's not divisible by 2).
We can conclude that 6501 is a prime number because other than 1 and 6501 itself, there are no other numbers that divide it evenly, indicating no divisors that give a whole number as the result.
The oblique pyramid has a square base. What is the volume of the pyramid? 2.5cm3 5cm3 6cm3 7.5cm3
Answer:
V = 5 cm³Step-by-step explanation:
The formula of a volume of a pyramid:
[tex]V=\dfrac{1}{3}BH[/tex]
B - base area
H - height
In the base we have a square withe side s = 2cm
The formula of an area of a square with side s:
[tex]A=s^2[/tex]
Substitute:
[tex]A=2^2=4\ cm^2[/tex]
The height H = 3.75 cm.
Calculate the volume:
[tex]V=\dfrac{1}{3}(4)(3.75)=\dfrac{15}{3}=5\ cm^3[/tex]
Answer: C. 58 1/3 cm∧2
Just took quiz.
The circle below is centered at the point (4,3) and has a radius of length 5.
What is its equation?
A. (x-4)2 + (y + 3)2 =
52
B. (x-4)2 + (y - 3)2 = 52
C. (x-3)2 + (y - 3)2 =
52
D. (x+4)2 + (y-3)2 =
25
Answer:
B. (x-4)^2 + (y - 3)^2 = 5^2
Step-by-step explanation:
The equation for a circle is given by
(x-h)^2 + (y-k) ^2 = r^2
where (h,k) is the center and r is the radius
We have a center of (4,3) and a radius of 5
(x-4)^2 + (y-3) ^2 = 5^2
The length of a rectangle is three times its width, and its area is 9 cm2
. Find the
dimensions of the rectangle.
Answer:
3√3 = l
√3 = w
Step-by-step explanation:
l = 3w
9 = 3w[w] ↷
9 = 3w² [Divide by 3]
3 = w² [Take the square root]
√3 = w [plug this back into the top equation to get a length of 3√3]
[3√3][√3] = 9 [Area]
I am joyous to assist you anytime.
A person's systolic blood pressure, which is measured in millimeters of mercury (mm Hg), depends on a person's age, in years. The equation:
P=0.005y^2−0.01y+121
gives a person's blood pressure, P, at age y years.
A.) Find the systolic pressure, to the nearest tenth of a millimeter, for a person of age 48 years.
B.) If a person's systolic pressure is 133 mm Hg, what is their age (rounded to the nearest whole year)?
Answer:
a) 132.0 mmHg, b) 50 years old.
Step-by-step explanation:
a) Plug in 48 where you see the letter y and simplify, preferably with a calculator.
P = 0.005(48)^2 - 0.01(48) + 121
P = 132.04 mmHg, to the nearest tenth would be 132.0 mmHg
b) Plug in 133 for P and solve for y.
0.005y^2 - 0.01 + 121 = 133
To make it a little easier on myself -- and because I haven't practiced a diff. method in a while -- I simplified the equation to 0.005y^2 - 0.01y - 12 = 0 by subtracting 133 from both sides. I did that so that I can could then use the quadratic formula to solve.
Quadratic formula is y = (-b +/- √(b^2 - 4ac)) / 2
Now we plug in our given information, that new trinomial, to solve for y
[tex]y = \frac{0.01 +/- \sqrt{(0.01)^2 - 4(0.005)(-12)} }{2(.0.005)} \\y = \frac{0.01 +/- \sqrt{0.2401}}{0.01}[/tex]
[tex]y = \frac{0.01}{0.01} +/- \frac{\sqrt{0.2401}}{0.01} \\y = 1 +/- \frac{0.49}{0.01}\\y = 1+/- 49[/tex]
Because it is a trinomial, you are given two answers. You get y = 48 and y = 50. In order to find out which is right, you plug in and see which on yields 133 as the answer. Given the part a), I already know it's not 48. When I plug in 50, I get 133. Therefore, 50 years old is your answer.
The systolic pressure for a person who is 48 years old is approximately 132.0 mm Hg. If a person's systolic pressure is 133 mm Hg, their age is approximately 49 years when rounded to the nearest whole year.
A) Systolic Pressure Calculation for Age 48
To find the systolic pressure for a person who is 48 years old, we use the given equation:
P = 0.005y^2 - 0.01y + 121
Substitute y = 48 into the equation:
P = 0.005(48)^2 - 0.01(48) + 121 = 0.005(2304) - 0.48 + 121 = 11.52 - 0.48 + 121 = 11.04 + 121 = 132.04 mm Hg
To the nearest tenth, the systolic pressure is 132.0 mm Hg.
B) Age Calculation for Systolic Pressure of 133 mm Hg
To find the age when a person's systolic pressure is 133 mm Hg, we set P = 133 and solve for y:
0.005y^2 - 0.01y + 121 = 133 0.005y^2 - 0.01y - 12 = 0
Using the quadratic formula, y = [-b ± √(b^2 - 4ac)] / (2a), where:
a = 0.005,
b = -0.01, and
c = -12.
We find the positive root that makes physical sense for age:
y ≈ 48.9 years
The age rounded to the nearest whole year is 49 years.
Order these numbers from least to greatest.
3/4, -1/5, -5/16, 0.90, -0.52
0.90,34 , -5/16, -0.52, -1/5
-1/5, -5/16, -0.52, 3/4, 0.90
0.90, 3/4, -0.52, -5/16, -1/5
-0.52, -5/16, -1/5, ,3/4 0.90
The numbers ordered from least to greatest are -0.52, -5/16, -1/5, 3/4, and 0.90. Negative values are ordered by ascending absolute value, while positive values are ordered normally.
Explanation:To order the numbers from least to greatest, we first need to compare the negative numbers, then the positive fractions and decimals. It's important to understand that negative numbers are less than zero, and the number with the largest absolute value is actually the smallest when negative. Positive numbers are greater than zero, with larger decimal or fractional values representing larger numbers.
-0.52 (because it is the only number less than -0.5)-5/16 (which is equal to -0.3125, so it's greater than -0.52 but still negative)-1/5 (or -0.2, which is the largest of the negative numbers)3/4 (equal to 0.75 and is less than 0.90)0.90 (as it is the greatest positive decimal given)The correct order from least to greatest is: -0.52, -5/16, -1/5, 3/4, and 0.90.
Alice placed 3 balls on a straight line at the points P (-5,9). Q (-5.-2), and R (-5.-9). Which two balls are separated by a
longer distance?
Check the picture below.
The greatest distance between the balls placed by Alice on a straight line is between the balls at points P and R, which equates to 18 units.
Explanation:The point coordinates given for the balls that Alice placed are located on a straight line. They are vertically aligned, as the x-coordinate is the same (-5) for all the balls. The y-coordinate represents the vertical position of each ball.
To find the greatest distance between the balls, calculate the distance between the highest point P (-5,9) and the lowest point R (-5,-9). The formula to calculate the distance between two points (x1, y1) and (x2, y2) on a straight line is sqrt[(x2 - x1)^2 + (y2 - y1)^2].
However, as x1 and x2 are the same, the calculation simplifies to |y2 - y1|. So the distance between P and R would be |-9 - 9| which equals 18 units. Therefore, the two balls at points P and R are separated by the greatest distance.
Learn more about Distance Measurement here:https://brainly.com/question/34491476
#SPJ2
On the first day of vacation, you read one-quarter of a novel. On the second day, you read half of the remaining pages. On the third day, you read the last 123 pages of the novel.
(a) How many pages does the novel have?
pages
(b) How many pages did you read by the end of the second day?
pages
Answer:
328 pages for the novel
123 pages for the second day
Step-by-step explanation:
Let the number of pages of the novel = x
Raw Equation
(1/4)x + 1/2 (3/4)x + 123 = x
Solution
(1/4)x + (3/8)x+ 123 = x
Change the fractions to common denominators.
(2/8)x + (3/8)x + 123 = x
Add the fractions.
(2/8 + 3/8)x + 123 = x
Subtract (5/8)x from both sides.
(5/8)x + 123 = x
(5/8)x- (5/8)x + 123 = x - (5/8)x
Multiply both sides by 8
123 = (3/8)x
123 * 8 = 3x
Divide by 3
984 = 3x
984/3 = 3x / 3
328 = x
===================
At the end of the second day, she read 3/8 * 328 = 123 pages.
The novel has 328 pages. By the end of the second day, the student had read 205 pages of the novel.
Explanation:The student's schoolwork question can be addressed by setting up and solving algebraic equations. Let's denote the total number of pages in the novel as x. On the first day, one-quarter of the novel is read, which is x/4 pages. So, there are 3x/4 pages remaining. On the second day, half of the remaining pages are read, which is (1/2) × (3x/4) = 3x/8 pages. On the third day, the student reads the last 123 pages, which were all the pages that were left. Therefore, the equation to solve for x is:
x - (x/4 + 3x/8) = 123
We can solve this equation to find out the total number of pages in the novel:
First, let's find a common denominator for the fractions. It is 8.
8x/8 - (2x/8 + 3x/8) = 123
8x/8 - 5x/8 = 123
3x/8 = 123
Let's multiply both sides of the equation by 8/3 to solve for x.
x = 123 × (8/3)
x = 328
The novel has 328 pages.
To find out how many pages were read by the end of the second day, we add the amount read on the first and second days:
(x/4) + (3x/8) = (2x/8) + (3x/8) = 5x/8
5x/8 when x = 328 is:
(5 × 328)/8 = 205
By the end of the second day, 205 pages were read.
A right pyramid with a square base has a base edge length of 12 meters and slant height of 6 meters.
The apothem is meters.
The hypotenuse of ΔABC is the .
The height is meters.
The volume of the pyramid is cubic meters.
Answer:
Step-by-step explanation:
Square base of the base edge = 12 meters and slant height = 6√2 meters
Apothem of the right pyramid will be = AC
1). Now AC = [tex]\sqrt{AB^{2}-BC^{2}}[/tex]
AC = [tex]\sqrt{(6\sqrt{2})^{2}-(6)^{2}}[/tex]
= [tex]\sqrt{72-36}=\sqrt{36}[/tex]
= 6
Now Apothem = 6 meters
2). Hypotenuse of Δ ABC = AB = 6√2 meters
3). Height AC = 6 meters
4). Volume of the pyramid = [tex]\frac{1}{3}(\text{Area of the base})(\text{Apothem})[/tex]
Volume = [tex]\frac{1}{3}(12)^{2}(6)[/tex]
= 2×144
= 288 meter²
Correct responses:
The apothem is 6 metersThe hypotenuse of ΔABC is 6·√2 metersThe height is 6 meterThe volume of the pyramid is 288 cubic metersMethods used for the calculationsThe given dimensions of the right pyramid having a square base are;
Base edge length, l = 12 meters
Slant height = 6·√3
Required:
Length of the apothem.
Solution:
The apothem, a, is the line drawn from the middle of the polygon to the midpoint of a side.
Therefore;
The apothem of the square base = [tex]\dfrac{l}{2} [/tex] = [tex]\overline{BC}[/tex]
Which gives;
[tex]a = \dfrac{12}{2} = 6[/tex]
The apothem is 6 metersRequired:
The hypotenuse of triangle ΔABC
Solution:
The hypotenuse of ΔABC = The slant height of the square pyramid = 6·√2 meters
Therefore;
The hypotenuse of ΔABC = 6·√2 metersRequired:
The height of the pyramid
Solution:
The height of the pyramid = The length of the side [tex]\overline{AC}[/tex] in right triangle
ΔABC, therefore, by Pythagorean theorem, we have;
[tex]\overline{AC}^2[/tex] = [tex]\overline{AB}^2[/tex] - [tex]\overline{BC}^2[/tex]
Which gives;
[tex]\overline{AC}^2[/tex] = (6·√2)² - 6² = 6²·((√2)² - 1) = 6² × 1 = 6²
[tex]\overline{AC}[/tex] = √(6²) = 6
The height of the pyramid, h = [tex]\overline{AC}[/tex] = 6 metersRequired:
The volume of the pyramid.
Solution:
[tex]The \ volume \ of \ a \ pyramid \ is, \ V = \mathbf{\dfrac{1}{3} \times Base \ area \times Height}[/tex]
[tex]V = \dfrac{1}{3} \times A \times h [/tex]
The base area of the square pyramid, A = 12 m × 12 m = 144 m²
Therefore;
[tex]V = \dfrac{1}{3} \times 144\, m^2 \times 6 \, m = 288 \, m^2[/tex]
The volume of the pyramid is V = 288 cubic metersLearn more about the volume of solids here:
https://brainly.com/question/13338580
simplify this expression
4^8 / 4^4
A. 2
B. 16
C. 64
D. 256
Answer:
[tex]\displaystyle 256[/tex]
Step-by-step explanation:
PEMDAS
P-parenthesis, E-exponent, M-multiply, D-divide, A-add, and S-subtracting.
Exponent rule: [tex]4^8^-^4[/tex]
Subtract by the exponent from left to right.
[tex]\displaystyle 8-4=4[/tex]
[tex]\displaystyle 4^4=4*4*4*4=256[/tex]
256 is the correct answer.
Can you please mildly explain this with the answer.
[tex]\bf \begin{cases} f(x)=&5x-1\\ f(-3)=&5(-3)-1\\ f(-3)=&-15-1\\ f(-3)=&-16 \end{cases}\qquad \begin{cases} g(x)=&2x^2+1\\ g(-3)=&2(-3)^2+1\\ g(-3)=&2(-3)(-3)+1\\ g(-3)=&2(9)+1\\ g(-3)=&18+1\\ g(-3)=&19 \end{cases} \\\\[-0.35em] ~\dotfill\\\\ (f\times g)(-3)\implies f(-3)\times g(-3)\implies (-16)(19)\implies -304[/tex]
A man is four times as older as his son. In four years time he will be three times as old. What are their ages now?
Answer:
The son is 8 years old
The father is 32 years old
Step-by-step explanation:
Let the man age be x
Let the son age = y
Right now the man is four times as older as his son = x=4y
In four years time he will be three times as old. ⇒x+4 =3(y+4)
Now substitute the value x=4y in x+4=3(y+4)
x+4=3(y+4)
4y+4=3(y+4)
4y+4=3y+12
Combine the like terms:
4y-3y =12-4
y=8
If the son is 8 years old than;
x=4y
x=4(8)
x=32
Father will be 32 years old....
Answer:
Present age of son: 8 years.
Present age of father: 32 years.
Step-by-step explanation:
Let x represent present age of the son and y represent present age of father.
We have been given that a man is four times as older as his son. 4 times of age of son would be [tex]4x[/tex].
We can represent this information in an equation as:
[tex]y=4x[/tex]
We are also told that in four years time he will be three times as old.
Age of father in 4 years would be [tex]y+4[/tex].
We can represent this information in an equation as:
[tex]y+4=3(x+4)[/tex]
Upon substituting [tex]y=4x[/tex] in 2nd equation, we will get:
[tex]4x+4=3(x+4)[/tex]
[tex]4x+4=3x+12[/tex]
[tex]4x+4-4=3x+12-4[/tex]
[tex]4x=3x+8[/tex]
[tex]4x-3x=3x-3x+8[/tex]
[tex]x=8[/tex]
Therefore, the present age of son is 8 years.
Upon substituting [tex]x=8[/tex] in equation [tex]y=4x[/tex], we will get:
[tex]y=4x\Rightarrow 4(8)=32[/tex]
Therefore, the present age of father is 32 years.
29 POINTS! ANSWER ASAP PLEASE. Which statements about the dilation are true? Check all that apply. (multiple choice question) (image provided below)
A. The center of dilation is point C
B. It is a reduction
C. It is an enlargement
D. The scale factor is 2.5
E. The scale factor is 2/5
Answer:
A. The center of dilation is point C.
B. It is a reduction.
E. The scale factor is 2/5.
Step-by-step explanation:
The center is shown as C. You can see this from the line segments they drew through C, the image, and the pre-image.
The pre-image is the image before the dialation. The pre-image here is XYZ.
The image is the image after dialation. The image is X'Y'Z'.
If you look at the pre-image XYZ and then it's image X'Y'Z', ask yourself the image get smaller or larger. To me I see a larger triangle being reduce to a smaller triangle so this is a reduction.
The scale factor cannot be bigger than 1 because the image shrunk so D is definitely not a possibility.
E. is a possibility but let's actually find the scale factor to see.
We can calculate [tex]\frac{CX'}{CX}[/tex] or [tex]\frac{CY'}{CY}[/tex] or [tex]\frac{CZ'}{CZ}[/tex] to find out what the scale factor is.
[tex]\frac{CX'}{CX}=\frac{2}{5}[/tex]
[tex]\frac{CZ'}{CZ}=\frac{3}{7.5}=\frac{2}{5}[/tex].
The scale factor is 2/5.
Answer:
A, The center of dilation is point C.
B, It is a reduction.
and E, The scale factor is 2/5.
Step-by-step explanation:
Can someone show me how to do the problem ? If I take 8 pills a week, how many months is that?
Answer:
32
Step-by-step explanation:
I assume you want to find the pills per month? You know that you take 8 pills per week, and there are 4 weeks in a month. So:
8 pills/week × (4 week / month) = 32 pills/month
Which expression represents the number rewritten in a + bi form?
Answer:
3 + 2i
Step-by-step explanation:
3 + √(-4) = 3 + i√4 = 3 + 2i
Recall that √(-1) = i
Please help!
Mrs. G is planning an expansion of her square back yard. If each side of the original backyard is increased by 5 m, the new total area of the backyard will be 196 m2. Find the length of each side of the original backyard.
Answer:
9 m
Step-by-step explanation:
If x is the length of the original square, then x+5 is the length of the new square.
Area of a square is the square of the side length, A = s². Since the area of the new square is 196 m²:
196 = (x + 5)²
Solving, first take the square root of both sides:
±14 = x + 5
Subtract 5 from both sides:
x = -5 ± 14
x = -19, x = 9
x can't be negative, so x = 9. The side length of the original backyard is 9 m.
Answer:
9m
Step-by-step explanation:
If Mrs. G is planning an expansion of her square back yard and side of the original backyard is increased by 5 m, the new total area of the backyard will be 196 m2. The length of each side of the original backyard is 9m.
x = length of original square
x + 5
Formula: A = s²
Therefore, the side length of the original backyard is 9 m.
simplify each of the following exponential expressions:
a.
[tex] {7}^{2} [/tex]
b.
[tex] {6}^{3} [/tex]
c.
[tex] {12}^{2} [/tex]
d.
[tex] {2}^{4} [/tex]
e.
[tex] {10}^{3} [/tex]
Answer:
the answer is A
Step-by-step explanation:
Two planes travel toward each other from cities that are about 450 km apart at rates of 240 km/hr and 210 km/hr. They started at the same time. In how many hours will they meet?
Answer:
1 hour.
Step-by-step explanation:
You have to do the total distance /total speed together.
You have to do this to ensure that you are counting how fast they are travelling together.
(450 km) / the sum of their speed (240+210)
450/ (240+210)
=450/ 450
=1
Answer: 1 Hour.
Answer:
t = 1 hour
Step-by-step explanation:
Oddly the distances add. Each plane will contribute a certain distance to make the 450 km. They will meet after the same number of hours have passed.
d = r * t
r1 = 240 km/hour
t1 = t
r2 = 210 km/hour
t2 = t
240*t + 210*t = 450 collect the terms on the left.
450t = 450 divide by 450
450t/450 = 450/450
t =1
After 1 hour they will meet.
Three times a number added twice a smaller number is 4. Twice the smaller number less than twice the larger number is 6. Find the number
Answer:
x= -2/5 and y=13/5
Step-by-step explanation:
Lets assume that the larger number = x
And the smaller number = y
According to the given statement three times a number added twice a smaller number is 4, it means;
3x+2y=4 -------- equation 1
Now further twice the smaller number less than twice the larger number is 6,it means;
2y-2x=6 --------equation 2
Solve the equation 2.
2y=2x+6
y=2x+6/2
y=2(x+3)/2
y=x+3
Substitute the value of y=x+3 in the first equation.
3x+2y=4
3x+2(x+3)=4
3x+2x+6=4
Combine the like terms:
5x=4-6
5x=-2
x= -2/5
Put the value x= -2/5 in equation 2.
2y-2x=6
2y-2(-2/5)=6
2y+4/5=6
By taking L.C.M we get
10y+4/5=6
10y+4=6*5
10y+4=30
10y=30-4
10y=26
y=26/10
y=13/5
Hence x= -2/5 and y=13/5....
To find the numbers, we first solve for x and y using a system of equations. We find that x is 2 and y is -1. Thus, the larger number is 2 and the smaller number is -1.
Detailed Explanation is as follows:
Let the larger number be x and the smaller number be y. Based on the problem, we can set up the following system of equations:
3x + 2y = 4
2x - 2y = 6
Add the equations together to eliminate y.
3x + 2y + 2x - 2y = 4 + 6
5x = 10
x = 2
Now, Substitute x back into one of the original equations
Using the first equation:
3(2) + 2y = 4
6 + 2y = 4
2y = -2
y = -1
Therefore, the larger number is 2 and the smaller number is -1.
Hence the larger number is 2 and the smaller number is -1.
Nathan has 102 solid-colored disks that are red.
blue, and green. He lines them up on the floor
and finds that there are 3 more red disks than blue
and 6 more blue disks than green. How many red
disks are there?
Answer:
blue = 35
green = 29
red = 38
Step-by-step explanation:
Let r = red
Let g = green
let b = blue
r + g + b = 102
r = b + 3
b = g + 6 Subtract 6 from both sides of the equation
b - 6 = g
===================
substitute for red and green
(b + 3) + b + b - 6 = 102
b + 3 + b + b - 6 = 102
Combine like terms
3b - 3 = 102
Add 3 to both sides
3b - 3 + 3 = 102 + 3
3b = 105
Divide by 3
3b/3 = 105/3
b = 35
======================
r = b + 3
r = 35 + 3
r = 38
=======================
g = b - 6
g = 35 - 6
g = 29
========================
Answer:
red=35 green=29 blue=38
Step-by-step explanation:
a baseball diamond is actually a square with 90 foot sides. If a number tries to steal second base. how far must the catcher, at home plate thrown to get the number out
Answer:
The catcher will have to throw roughly 127 feet from home to second base.
Step-by-step explanation:
We have to use the Pythagorean Theorem. This gives us the third side length if we know the other two side lengths.
a² + b² = c²
90² + 90² = c²
8100 + 8100 = c²
16200 = c²
[tex]\sqrt{16200}[/tex] = [tex]\sqrt{c^2}[/tex]
c = 127.28 ft
plz show that
[tex] \tan( \frac{\pi}{4} - \alpha ) \: \tan( \frac{\pi}{4} + \alpha ) = 1[/tex]
Solution:
The formula for tan(A+B) and tan(A-B) are:
[tex]tan(A+B) = \frac{tan(A)+tan(B)}{1-tan(A)tan(B)} \\\\tan(A-B) = \frac{tan(A)-tan(B)}{1+tan(A)tan(B)}[/tex]
The left hand side of the given expression is:
[tex]tan(\frac{\pi}{4}-\alpha ) tan(\frac{\pi}{4}+\alpha )[/tex]
Using the formula above and value of tan(π/4) = 1, we can expand this expression as:
[tex]tan(\frac{\pi}{4}-\alpha ) tan(\frac{\pi}{4}+\alpha )\\\\ = \frac{tan(\frac{\pi}{4} )-tan(\alpha)}{1+tan(\frac{\pi}{4} )tan(\alpha)} \times \frac{tan(\frac{\pi}{4} )+tan(\alpha)}{1-tan(\frac{\pi}{4} )tan(\alpha)}\\\\ = \frac{1-tan(\alpha)}{1+tan(\alpha)} \times \frac{1+tan(\alpha)}{1-tan(\alpha)}\\\\ = 1 \\\\ = R.H.S[/tex]
Thus, the left hand side is proved to be equal to right hand side.
4.
Vivian has some sweets. If she shares the sweets among 4 friends, she will have
3 sweets left. If she shares the sweets among 5 friends, she will have
4 sweets left. If she shares the sweets among 9 friends, she will have
8 sweets left. What is the smallest possible number of sweets she has?
What is the solution to this question?
Answer:
least possible number of sweets = lowest common multiple of 5,6 & 10 - 2
-I hope this helps! I got it figured out until near like the very end.-
-Please mark as brainliest!- Thanks!
Final answer:
The smallest number of sweets that satisfies the condition of being left with certain remainders when shared with different numbers of friends is found using modular equations, with the solution being 59 sweets.
Explanation:
To solve the problem presented for Vivian and her sweets, we will use the concept of simultaneous congruences from number theory.
Vivian's sweets when divided by 4 leave a remainder of 3, when divided by 5 leave a remainder of 4, and when divided by 9 leave a remainder of 8. This situation translates to the following set of modular equations:
Sweets ≡ 3 (mod 4)Sweets ≡ 4 (mod 5)Sweets ≡ 8 (mod 9)The smallest number that satisfies all these conditions is known as the least common multiple plus the respective remainders.
With the aid of the Chinese Remainder Theorem, we can conclude that the smallest number of sweets that satisfies all conditions is 59. This is the least number of sweets Vivian can have and still meet the conditions given for sharing among her friends.
to the nearest hundreth, what is the circumference of a circle with a radius of 7 units?
Answer:153.94
Step-by-step explanation:circumference is πr^2
π × 49 = 153.93804
Nearest hundredth count two digits after the decimal(to make an hundredth), then round off with the third number
If p represents a digit and 4p/6p + 5/17 = 1, what digit does p represented?
Answer:
p=8
Step-by-step explanation:
We know that 'p' represents a digit.
We can say that:
x + 5/17 = 1
Solving for 'x' we have:
x = 1 -5/17
x = 12/17
Now, multiplying 'x' by 4, we have:
x = 48/68
Therefore, p=8
which formula can be used to describe the sequence below 27,9,3
Answer:
[tex]a_n=27 \cdot (\frac{1}{3})^{n-1}[/tex].
Step-by-step explanation:
This is a geometric sequence that means there is a common ratio. That means there is a number you can multiply over and over to get the next term.
The first term is 27.
The second term is (1/3)(27)=9.
The third term is (1/3)(9)=3.
So the common ratio is 1/3.
That means you can keep multiplying by 1/3 to find the next term in the sequence.
The explicit form for a geometric sequence is [tex]a_n=a_1 \cdot r^{n-1}[/tex] where [tex]a_1[/tex] is the first term and [tex]r[/tex] is the common ratio.
We are given [tex]a_1=27[/tex] and [tex]r=\frac{1}{3}[/tex].
So the explicit form for the given sequence is [tex]a_n=27 \cdot (\frac{1}{3})^{n-1}[/tex].
Answer:
Its b on edge 2021
Step-by-step explanation:
just did it
the figure (3,12,9,3) contains only horizontal and vertical lines. Calculate its perimeter.
Answer:
30 square units
Step-by-step explananation:
First of all we need to know the formula to finding the perimeter of a rectangle which is:
P = 2 x L(Length) + 2 x h(Height)
12 and 3 are apart by 12 - 3
12 - 3 = 9
Then, we subtract 9 from 3 ( :
9 - 3 = 6)
To get 6 as our answer.
6 will be the width and 9 will be the length.
Now we solve for the perimeter by plugging in our values into our formula:
P = 2(9) + 2(6)
P = 18 + 12
P = 30 square units
1/4 x ≤ -3 solve for x
moving 4 to the other side x= 4 × -3 =-12
so x =-12
Answer:
x ≤ -12
Step-by-step explanation:
1/4 x ≤ -3
Multiply by 4 on both sides to get rid of the fraction coefficient.
x ≤ -12
This means x is x is equal to or less than -12
cos pi/4 cos pi/6= 1/2(___pi/12+cos 5pi/12) fill in the blank
Answer:
[tex]\cos(\frac{\pi}{4})\cos(\frac{\pi}{6})=\frac{1}{2}(\cos(\frac{\pi}{12})+\cos(\frac{5\pi}{12}))[/tex]
So the blank is cos.
Step-by-step explanation:
There is an identity for this:
[tex]\cos(a)\cos(b)=\frac{1}{2}(\cos(a+b)+\cos(a-b))[/tex]
Let's see if this is fit by your left hand and right hand side:
So [tex]a=\frac{\pi}{4}[/tex] while [tex]b=\frac{pi}{6}[/tex].
Let's plug these in to the identity above:
[tex]\cos(\frac{\pi}{4})\cos(\frac{\pi}{6})=\frac{1}{2}(\cos(\frac{\pi}{4}+\frac{\pi}{6})+\cos(\frac{\pi}{4}-\frac{\pi}{6}))[/tex]
Ok, we definitely have the left hand sides are the same.
Let's see if the right hand sides are the same.
Before we move on let's see if we can find the sum and difference of [tex]\frac{\pi}{4}[/tex] and [tex]\frac{\pi}{6}[/tex].
We will need a common denominator. How about 12? 12 works because 4 and 6 go into 12. That is 4(3)=12 and 6(2)=12.
[tex]\frac{\pi}{4}+\frac{\pi}{6}=\frac{3\pi}{12}+\frac{2\pi}{12}=\frac{5\pi}{12}[/tex].
[tex]\frac{\pi}{4}-\frac{\pi}{6}=\frac{3\pi}{12}-\frac{2\pi}{12}=\frac{\pi}{12}[/tex].
Let's go back to our identity now:
[tex]\cos(\frac{\pi}{4})\cos(\frac{\pi}{6})=\frac{1}{2}(\cos(\frac{\pi}{4}+\frac{\pi}{6})+\cos(\frac{\pi}{4}-\frac{\pi}{6}))[/tex]
[tex]\cos(\frac{\pi}{4})\cos(\frac{\pi}{6})=\frac{1}{2}(\cos(\frac{5\pi}{12})+\cos(\frac{\pi}{12}))[/tex]
We can rearrange the right hand side inside the ( ) using commutative property of addition:
[tex]\cos(\frac{\pi}{4})\cos(\frac{\pi}{6})=\frac{1}{2}(\cos(\frac{\pi}{12})+\cos(\frac{5\pi}{12}))[/tex]
So comparing my left hand side to their left hand side we see that the blank should be cos.
This is a trigonometric equation requiring the use of sum-to-product identities. By applying the identity and simplifying, the result is: cos(pi/4)cos(pi/6) = 1/2(cos pi/12 + cos 5pi/12). The blank should be filled by a '+' sign.
Explanation:The question refers to a trigonometric identity equation in the field of mathematics. To solve it, we have to use the principle of the sum-to-product identities from trigonometry.
Let's use the identity: cos(A)cos(B) = 1/2 [cos(A - B) + cos(A + B)]. In this case, A = pi/4 and B = pi/6.
Therefore, cos(pi/4)cos(pi/6) = 1/2 [cos(pi/4 - pi/6) + cos(pi/4 + pi/6)].
Simplified further cos(pi/24) + cos(5pi/12).
So, the blank should be filled with a '+', making the equation cos(pi/4)cos(pi/6) = 1/2(cos pi/12 + cos 5pi/12).
Learn more about trigonometry here:https://brainly.com/question/31896723
#SPJ11
Out of the 21 students in Mrs. Clark's class, 5of the class are boys and of the class are girls. How many students in the
class are girls and how many are boys?
(SHOW WORK)
Answer:
There are 4 boys and 17 girls in the class.
Step-by-step explanation:
Out of 21 students in Mrs. Clark's class, 5th of the students are boys and rest are girls.
out of 21 every 5th student is boy, so total boys are = [tex]\frac{21}{5}[/tex]
= [tex]4\frac{1}{5}[/tex]
The whole number is 4, so the number of boys are 4.
The number of girls = 21 - 4 = 17 girls
There are 4 boys and 17 girls in the class.