Answer:
-7/2 ±1/2sqrt(13) = x
Step-by-step explanation:
f(x) =x^2 + 7x + 9
To find the zeros, set this equal to zero
0 = x^2 + 7x + 9
I will complete the square
Subtract 9 from each side
0-9 = x^2 + 7x + 9-9
-9 =x^2 + 7x
Take the coefficient of the x term, 7
divide by 2, 7/2
Then square it, (7/2)^2 = 49/4
Add this to both sides
-9 +49/4=x^2 + 7x + 49/4
-36/4 +49/4 = (x+7/2)^2
13/4 = (x+7/2)^2
Take the square root of each side
±sqrt(13/4) = sqrt( (x+7/2)^2)
± sqrt(13) /sqrt(4)= (x+7/2)
± 1/2 sqrt(13) = (x+7/2)
Subtract 7/2 from each side
-7/2 ±1/2sqrt(13) = x+7/2-7/2
-7/2 ±1/2sqrt(13) = x
The function f(x) = x^2 + 7x + 9 has no real-number zeros as the discriminant is negative, indicating that the quadratic formula solution involves an imaginary number.
Explanation:The student is asking to find the zeros of the quadratic function f(x) = x^2 + 7x + 9. To solve for the zeros, we need to find the values of x that make the function equal to zero. We can use the quadratic formula, which is [tex]x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{{2a}}[/tex], where a, b, and c are the coefficients of the quadratic equation ax^2 + bx + c = 0.
In this case, our equation is already in the correct form with a = 1, b = 7, and c = 9. Plugging these into the quadratic formula, we get:
[tex]x = \frac{{-7 \pm \sqrt{{7^2 - 4(1)(9)}}}}{{2 \cdot 1}}[/tex]
Upon further calculation, we find that the equation has no real-number solutions as the discriminant (b^2 - 4ac) is negative (49 - 36 = 13), leading to an imaginary number in the square root. Therefore, we conclude that the function does not cross the x-axis and has no zeros on the real number line.
What is the simplified form of the following expression? 2sqrt18+3sqrt2+sqrt162 HURRY PLEASE HELP
Answer:
[tex]18\sqrt{2}[/tex]
Step-by-step explanation:
We need to simplify the following expression: [tex]2\sqrt{18} + 3\sqrt{2} + \sqrt{162}[/tex]
Then:
[tex]6\sqrt{2} + 3\sqrt{2} + 9\sqrt{2}[/tex]
[tex]18\sqrt{2}[/tex]
Therefore, the result is: [tex]18\sqrt{2}[/tex]
Complete the table for the given rule y=x-3
For the first row, where x is equal to 7, to find y plug 7 in for x like so...
y = 7 - 3
y = 4
For the second row, where y is equal to 1, to find x plug 1 in for y like so...
1 = x - 3
To solve for x add 3 to both sides. This will cancel 3 from the right side:
1 + 3 = x - 3 + 3
4 = x + 0
x = 4
For the third row, where y is equal to 7, to find x plug 7 in for y like so...
7 = x - 3
To solve for x add 3 to both sides. This will cancel 3 from the right side:
7 + 3 = x - 3 + 3
10 = x + 0
x = 10
First row: y is 4
Second row: x is 4
Third row: x is 10
Hope this helped!
~Just a girl in love with Shawn Mendes
The data shown on the scatter plot below demonstrates the relationship between a young boy's age (in months) and the
average number of hours that he sleeps each night.
The slope of the best fit line shows that as the boy's age _________,
the average number of hours
that he sleeps each night _________.
Blank A options:
Decreases
Increases
Stays the same
Blank B options:
Increases
Decreases
Stays the same
Answer:
a) increases b) decreases
Step-by-step explanation:
Typically, descriptions of trends are indicated as the change of the dependent (y) variable with respect to the increase of the independent (x) variable. In this case, age is the x variable and hours of sleep is the y variable.
Answer:
The data shown on the scatter plot below demonstrates the relationship between a young boy’s age (in months) and the average number of hours that he sleeps each night.
Step-by-step explanation:
The slope of the best fit line shows that as the boy’s age
increases
the average number of hours that he sleeps each night
decreases
.
Write an equation for a rational function with:
Vertical asymptotes of x = 2 and x = -3
x intercepts of (-6,0) and (3,0)
Horizontal asymptote of y = 6
Use y as your output variable.
Answer:
y=(6(x+6)(x-3))/((x-2)(x+3))
Step-by-step explanation:
The vertical asymptote should be in the denominator. The x-interceps should be in the numerator. Because we have horizontal asymptote y=6, then we have to put 6 in the numerator. the horizontal asymptote is the leading coefficient of the numerator ÷ the leading coefficient of the denominator, when the degree of the numerator and denominator are the same.
Given the horizontal asymptote, vertical asymptotes and x intercepts, the equation of the rational function is y = 6((x+6)(x-3))/((x-2)(x+3)). The vertical asymptotes are found by setting the function's denominator equal to zero, while the x-intercepts come from setting the numerator to zero.
Explanation:In this question, we are asked to write the equation of a rational function based on given conditions. The function's vertical asymptotes are located at x = 2 and x = -3, and has x-intercepts at (-6,0) and (3,0), with a horizontal asymptote at y = 6.
The general form of a rational function is y = (ax+b)/(cx+d). Asymptotes help define the behavior and boundaries of the function. In this situation, we can set the denominator of our function equal to zero to find our vertical asymptotes, giving us (x-2)(x+3). To achieve our stated x-intercepts, we set the numerator equal to zero, providing (x+6)(x-3). Combining these, the function becomes y = ((x+6)(x-3))/((x-2)(x+3)). The output of the function approaches the horizontal asymptote as x approaches infinity. Thus to have y = 6 as our horizontal asymptote, we adjust our function to maintain this behaviour, settling on y = 6((x+6)(x-3))/((x-2)(x+3)).
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Which of the equations below can be used to find the measure of ∠A?
A. A2=6.7^2+9.4^2
B. cosA=6.7/9.4
C. tanA=9.4/6.7
D. sinA/9.4=sin90/6.7
Answer:
I believe the answer is C.
The trigonometric function gives the ratio of different sides of a right-angle triangle. The correct option is C.
What are Trigonometric functions?The trigonometric function gives the ratio of different sides of a right-angle triangle.
[tex]\rm Sin \theta=\dfrac{Perpendicular}{Hypotenuse}\\\\\\Cos \theta=\dfrac{Base}{Hypotenuse}\\\\\\Tan \theta=\dfrac{Perpendicular}{Base}[/tex]
where perpendicular is the side of the triangle which is opposite to the angle, and the hypotenuse is the longest side of the triangle which is opposite to the 90° angle.
To find the measure of angle A, we need to use the tangent trigonometric function this is because except for the hypotenuse of the triangle the other two sides of the equation are known.
Therefore, the equation of the tangent function of trigonometry for angle A can be written as,
tan(A) = Perpendicular /Base
tan(A) = 9.4 / 6.7
Hence, the equations below that can be used to find the measure of ∠A is tanA=9.4/6.7.
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If 70% of a class is girls, and there are 30 students in the class; how many boys are in the class?
Answer:
9
Step-by-step explanation:
70% of 30 is 21, so to find the number of boys, take the original 30, and subtract 21, leaving you with 9, hope this helps and good luck bud :)
There are 9 boys in the class.
To determine the number of boys in a class where 70% are girls, multiply the total number of students by the percentage of boys in the class.
Given:
The class consists of 30 students with 70% being girls.To find the number of boys, we can calculate 30 students * 30% (100% - 70% girls) = 9 boys.Answer: There are 9 boys in the class.
PLEASE HELP AND ANSWER!!!!! Which of the following reveals the minimum value for the equation 2x2 + 12x − 14 = 0?
2(x + 6)2 = 26
2(x + 6)2 = 20
2(x + 3)2 = 32
2(x + 3)2 = 30
Answer:
The correct option is 3.
Step-by-step explanation:
The given equation is
[tex]2x^2+12x-14=0[/tex]
It can be written as
[tex](2x^2+12x)-14=0[/tex]
Taking out the common factor form the parenthesis.
[tex]2(x^2+6x)-14=0[/tex]
If an expression is defined as [tex]x^2+bx[/tex] then we add [tex](\frac{b}{2})^2[/tex] to make it perfect square.
In the above equation b=6.
Add and subtract 3^2 in the parenthesis.
[tex]2(x^2+6x+3^2-3^2)-14=0[/tex]
[tex]2(x^2+6x+3^2)-2(3^2)-14=0[/tex]
[tex]2(x+3)^2-18-14=0[/tex]
[tex]2(x+3)^2-32=0[/tex] .... (1)
Add 32 on both sides.
[tex]2(x+3)^2=32[/tex]
The vertex from of a parabola is
[tex]p(x)=a(x-h)^2+k[/tex] .... (2)
If a>0, then k is minimum value at x=h.
From (1) and (2) in is clear that a=2, h=-3 and k=-32. It means the minimum value is -32 at x=-3.
The equation [tex]2(x+3)^2=32[/tex] reveals the minimum value for the given equation.
Therefore the correct option is 3.
The correct answer is option 3. [tex]2(x + 3)^2 = 32[/tex].
To find the minimum value of the quadratic equation [tex]2x^2 + 12x - 14[/tex] = 0, we can rewrite it in vertex form, which reveals the minimum or maximum value of a quadratic function.
The given options are attempts at rewriting the quadratic equation in vertex form. Let’s rewrite the equation:
First, complete the square:
1. Start with the equation: [tex]2x^2 + 12x - 14[/tex]
2. Factor out the coefficient of x² from the first two terms: [tex]2(x^2 + 6x) - 14[/tex]
3. Complete the square inside the parentheses:
- Take [tex](\frac{6}{2})^2 =9[/tex] - Add and subtract 9 inside the parentheses: [tex]2(x^2 + 6x + 9 - 9) - 14[/tex]
- Simplify inside the square: [tex]2((x + 3)^2 - 9) - 14[/tex]
4. Distribute and simplify: [tex]2(x + 3)^2 - 18 - 14 = 2(x + 3)^2 - 32[/tex]
Comparing this with the options, we have [tex]2(x + 3)^2 = 32[/tex].
The correct answer is: [tex]2(x + 3)^2 = 32[/tex].
Type the correct answer in the box.
The formula for the volume, V, of a cone having the radius, r, and the height, h, is shown below.
V = žar24
Write the formula to calculate the height, h.
Answer:
[tex]\large\boxed{h=\dfrac{3V}{\pi r^2}}[/tex]
Step-by-step explanation:
[tex]\text{The formula of a volume of a cone:}\ V=\dfrac{1}{3}\pi r^2h.\\\\\text{Solve for}\ h:\\\\\dfrac{1}{3}\pi r^2h=V\qquad\text{multiply both sides by 3}\\\\3\!\!\!\!\diagup^1\cdot\dfrac{1}{3\!\!\!\!\diagup_1}\pi r^2h=3V\qquad\text{divide both sides by}\ \pi r^2\\\\h=\dfrac{3V}{\pi r^2}[/tex]
If point P is 4/7 of the distance from M to N, what ratio does the point P partition the directed line segment from M to N into?
4:1
4:3
4:7
4:10
Answer:
4:3
Step-by-step explanation:
Given that P divides segment MN into 4/7, let MN to be x units in length then
MP = 4/7 x =4x/7 --------(i)
But MN =MP+PN so;
x=4x/7 +PN
x- 4X/7 =PN
3x/7 =PN ----------(ii)
To get the ratio of MP:PN
MP: PN
4x/7:3x/7
MP/PN = 4x/7 / 3x/7
MP/PN =4/3
MP:PN = 4:3
Answer: 4:3
Step-by-step explanation:
Given : A point P is 4/7 of the distance from M to N.
∴ Let the distance between M to N be d.
[tex]\Rightarrow\ MP=\dfrac{4}{7}\times d=\dfrac{4d}{7}[/tex]
Also, the point P partition the directed line segment from M to N .
Thus , MN = MP+PN
[tex]\Rightarrow\ d=\dfrac{4d}{7}+PN\\\\\Rightarrow\ PN= d-\dfrac{4d}{7}=\dfrac{7d-4d}{7}\\\\\Rightarrow\ PN=\dfrac{3}{7}d[/tex]
Now, the ration of MP to PN will be :-
[tex]\dfrac{MP}{PN}=\dfrac{\dfrac{4d}{7}}{\dfrac{3d}{7}}=\dfrac{4}{3}[/tex]
∴ Point P partitioned the line segment MN into 4:3.
A square has sides of length 70 meters. What is the perimeter?
Answer:
280
Step-by-step explanation:
If a square has sides of length 70 meters, the perimeter is 280.
Formula: P=4a
P = 4a = 4·70 = 280
2/5 + 1/4 + 7/10 =
Answer:
27/20
Step-by-step explanation:
Adding Fractions7+1=8
5*4=20
5*1=5
8/20+5/20+14/20
Add numbers from left to right to find the answer.
8+5=13+14=27
27/20 is the correct answer.
ali and jake went on a cross country trip they took a train part of the way and took a bus the rest of the way they traveled a total of 1450 riding on the train 150 more kilometers than on the bus
let x=kilometers traveled by bus
let y = kilometers traveled by train
question how many kilometers did they travel by train?
Answer:
=800 km
Step-by-step explanation:
Let the distance traveled by train be y and by bus be x.
Bus -x
Train -y
y=x+150 (since they traveled by train for a distance of 150 km more than by bus.)
The sum of the two is equal to 1450
x+y=1450
y=x+150
These two form simultaneous equations.
y+x=1450..............i
y-x=150.................ii
Adding ii to i gives:
2y=1600
Divide both sides by two
y=800
Distance traveled by train =y=800 km
Answer:
800
Step-by-step explanation:
Write 0.68as a fraction in the lowest terms.
Answer:
17/25
Step-by-step explanation:
First, convert the decimal into a fraction. To do so, move the decimal point to the right two place values and place over 100.
0.68 = 68/100
Next, simplify. Divide common factors. Remember, what you do to one side, you do to the other. Divide 4 from both sides:
(68/100)/4 = (17/25)
17/25 is your answer.
~
0.68 can be expressed as the fraction 17/25 in the lowest terms.
A fraction in its lowest termStep 1: Let x be the decimal representation of the fraction.
x = 0.68
Step 2: Since there are two digits after the decimal point, we can multiply both sides of the equation by 100 to eliminate the decimal.
100x = 68
Step 3: Simplify the fraction by dividing both the numerator and denominator by their greatest common divisor (GCD), which in this case is 4.
100 ÷ 4 = 25
68 ÷ 4 = 17
The simplified fraction is:
0.68 = 17/25
Therefore, 0.68 can be expressed as the fraction 17/25 in the lowest terms.
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Which of the following is the correct graph of the compound inequality 4p + 1 > −15 or 6p + 3 < 45?
Answer:
4p + 1 > −15 or 6p + 3 < 45
has solution any number.
The graph looks like this
<~~~~~~~~~~~~~~~~~~~~~~~~~~~>
---------(-4)---------(7)-------------
The shading is everywhere from left to right.
Step-by-step explanation:
Let's solve this first:
4p+1>-15
Subtract 1 on both sides:
4p>-16
Divide both sides by 4:
p>-4
or
6p+3<45
Subtract 3 on both sides:
6p<42
Divide both sides by 6:
p<7
So our solution is p>-4 or p<7
So let's graph that
~~~~~~~~~~~~~~~~~~~~~~~~~~~~O
O~~~~~~~~~~~~~~~~~~~~~~~~~~~~ p>-4
---------------------(-4)---------------------(7)--------------------
or is a key word! or means wherever the shading exist for either is a solution.
So this shading is everywhere.
The answer is all real numbers.
The final graph looks like this:
<~~~~~~~~~~~~~~~~~~~~~~~~~~~>
---------(-4)---------(7)-------------
The shading is everywhere from left to right.
Answer:
Solution is (-∞,∞)
Step-by-step explanation:
[tex]4p + 1 > -15 \ or \ 6p + 3 < 45[/tex]
Solve each inequality separately
[tex]4p + 1 > -15[/tex]
Subtract 1 from both sides
[tex]4p> -16[/tex]
Divide both sides by 4
[tex]p> -4[/tex]
Solve the second inequality
[tex]6p + 3 < 45[/tex]
Subtract 3 from both sides
[tex]6p< 42[/tex]
Divide both sides by 6
[tex]p< 7[/tex]
[tex]p> -4 \ or \p< 7[/tex]
Solution is (-∞,∞)
Sara bought a soft drink for four dollars and five candy bars. She spent a total of nineteen
dollars. How much did each candy bar cost?
Answer:
The cost of each candy bar is $3....
Step-by-step explanation:
Let:
x= $4 (soft drink)
z= price of each candy bar
y=5z(Total price of 5 candy bars)
C= $19(spent money)
The equation is:
C= x+y
$19=$4+5z
Subtract 4 from both sides:
$15=5z
Divide both sides by 5
z=$3
Hence the cost of each candy bar is $3....
what is the slope intercept equation of the line below
Answer:
y=2x-3
Step-by-step explanation:
The slope-intercept form of a linear equation is y=mx+b where m is the slope and b is the y-intercept.
The y-intercept is where it crosses the y-axis. It cross the y-axis in your picture at -3 so b=-3.
Now the slope=rise/run. So starting at (0,-3) we need to find another point that crosses nicely on the cross-hairs and count the rise to and then the run to it. So I see (1,-1) laying nicely. So the rise is 2 and the run is 1.
If you don't like counting. You could just use the slope formula since we already identified the two points as (-1,1) and (0,-3).
The way I like to use the formula is line up the points and subtract vertically then put 2nd difference over 1st difference.
(0,-3)
-(1,-1)
----------
-1 -2
So the slope is -2/-1 or just 2.
We have that m is 2 and b is -3.
Plug them into y=mx+b and you are done.
y=2x-3.
Slope intercept equation of the line is y = 2x - 3.
What is slope-intercept form?Slope intercept form gives the graph of a straight line and is represented in the form of y=mx + c.
By checking the graph by drawing manually.
From that we get the equation
y = 2x - 3
Comparing above equation with the standard slope-intercept form y = mx +c, we get
Slope : m = 2
Now, given equation can be re-written as :
2x - y = 3
Divide by 3 on both sides
[tex]\frac{2x}{3} -\frac{y}{3} =\frac{3}{3}[/tex]
[tex]\frac{x}{\frac{3}{2} } -\frac{y}{3} =1[/tex]
Comparing above equation with intercept form:
[tex]\frac{x}{a}+\frac{y}{b}=1[/tex], we get
x-intercept : [tex]a=\frac{3}{2}[/tex]
y-intercept : [tex]b=-3[/tex]
Now the given straight line intersects the coordinate axes at [tex](\frac{3}{2} ,0)[/tex] and [tex](0,-3)[/tex]. Specify these plots on XY-plane & join by a straight line to get a plot.
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Jayla bought a table and some chairs at a furniture store.
The equation that models this situation is y = 35x+ 30, where y is the amount
of money she spent and x is the number of chairs she bought.
What does the y-intercept mean in this situation?
A. She paid $35 for each chair.
B. She paid $30 for each chair.
C. She paid $30 for the table.
D. She paid $35 for the table.
Answer:
C. She paid $30 for the table.
Step-by-step explanation:
y = 35x+ 30
This is in slope intercept form
y =mx+b
where m is the slope, which tells us the change
and b is the y intercept, which tells us how much the initial value was (when x=0)
Since x is the number of chairs, and letting x=0
y= 0 chairs +30
Since y= cost of the chairs and the table
y= 30, which is the cost of the table since we bought 0 chairs
which geometric object is defined as the set of ballpoints in a plane at a given distance A.Parabole B. Line segment C. Line D. circle
Answer:
Answer is Circle
Step-by-step explanation:
Check the picture below.
notice, all points are equidistant from the center of it, wherever the center happens to be.
Identify the restrictions on the domain of f(x) = quantity x plus 2 over quantity x minus 3.
For this case we must find the domain of the following function:
[tex]f (x) = \frac {x + 2} {x-3}[/tex]
By definition, the domain of a function is given by all the values for which the function is defined. The given function is not defined when the denominator is equal to zero, that is:
[tex]x-3 = 0\\x = 3[/tex]
Thus, the domain of the function is given by all real numbers except 3.
Answer:
x other than 3
Which is an equation of the line whose
slope is 2 and which passes through the
point (-2, 3)?
(1) y=-28+7 (3) y = -2x+1
(2) y = 2x +1 (4) y = 2x + 7
Please help!
Answer:
y = 2x + 7
Step-by-step explanation:
The equation of a line in slope- intercept form is
y = mx + c ( m is the slope and c the y- intercept )
Here m = 2, so
y = 2x + c ← is the partial equation
To find c substitute (- 2, 3 ) into the partial equation
3 = - 4 + c ⇒ c = 3 + 4 = 7
y = 2x + 7 ← equation of line
Need answer to A and B!
To find the probability of drawing certain colour of balls, we first need to find the total number of balls:
5 + 7 +5 = 17
For the number of red balls that need to be added
The original probability of drawing a red ball:
red ball/ total number of balls
= 5/17
To find the number of balls required, we can set an equation.
Let the number of red balls that need to be added be x.
5+x / 17+x = 5/6
(5+x) x 6 = (17+ x) x5
30 + 6x = 85 + 5x
6x - 5x = 85 - 30
x = 55
Therefore, 55 red balls need to be added.
For the number of black balls that need to be added
The original probability of drawing a white ball:
white ball/ total number of balls
=5/17
To find the probability required, we can set an equation.
Let the number of black balls that need to be added be y.
5/ 17 + y = 1/6
5 x 6 = 17 + y
30 = 17 + y
30 - 17 = y
y = 13
Therefore, 13 black balls need to be added.
Hope it helps!
Find the area of the figure. Round to the nearest tenth if necessary.
Question 2 options:
748
374
78
1496
Answer:
748 mm^2.
Step-by-step explanation:
We can split it up into 2 congruent triangles whose common base is 44 mm. and who have the same height 17 mm.
Area of a triangle= 1/2 * base * height.
So the area of the whole figure = 2 * 1/2 * 44 * 17.
= 44 *17
= 748 mm^2.
Answer:
It's 78 when you round it to nearest 10
IXL!! I WILL ADD BRAINLIST!
Answer:
-3,3.1, 3 2/10
Step-by-step explanation:
When you are putting different numbers from least to greatest, you need to put them in the same form...
First things first we should put -3 wayy up front. negative numbers are worth less than positive numbers.
Then we need to turn 3 2/10 into a decimal.
3 2/10 -------> 3.2
3.2 is more than 3.1
Thus, the order is
-3,3.1, 3 2/10
Hope this helps and have a nice day.
Answer:
-3, 3.1, 3 2/10
Step-by-step explanation:
3 2/10, 3.1, -3
Negative numbers are smaller than positive numbers
-3
Then we need to compare fractions and decimals.
Lets change the fraction to a decimal
2/10 = .2
3.2 and 3.1
3.1 < 3.2
So in order from least to greatest
-3, 3.1, 3 2/10
You are given a line that has a slope of 4 and passed through the point (3/8, 1/2). Which statements about the question of the line are true ? Check all that apply
Answer:
Statement 1, 2 and 4 are true where as statement 3 is not true.
Step-by-step explanation:
Statement 1: The y-intercept is -1
Point (3/8, 1/2)
Slope = m = 4
y = mx + c
1/2 = 4(3/8) + c
1/2 = 3/2 + c
1/2 = 3/2 + 2c/2
-2 = 2c
c = -1
This statement is true as the y-intercept is -1.
Statement 2: The slope intercept equation is y= 4x - 1
slope = m = 4
y-intercept = c = -1
y = mx + c
y = 4x - 1
This statement is true as the slope intercept equation is y= 4x -1
Statement 3: The point slope equation is y - 3/8 = 4 (x - 1/2)
Point slope equation: y - y1 = m (x - x1)
Points: (x, y) (3/8, 1/2)
y1 = 1/2
x1 = 3/8
Slope = m = 4
y - 1/2 = 4 (x - 3/8)
This statement is not true as the slope intercept equation is y - 1/2 = 4 (x - 3/8) instead of y - 3/8 = 4 (x - 1/2).
Statement 4: The point (3/8, 1/2) corresponds to (x1, y1) in the point slope form of the equation
This statement is true as shown in statement 3's explanation where x1 = 3/8 and y1 = 1/2
!!
How do you use theorems about triangles to solve problems?
Step-by-step explanation:
Triangles can be solved if you know either of three pieces of information:
Three sidesTwo sides and one angleTwo angles and one sideYou can solve for the remaining sides and angles using law of sine and law of cosine.
Law of sine:
(sin A) / a = (sin B) / b = (sin C) / c
Law of cosine:
c² = a² + b² − 2ab cos C
Here, A is the angle opposite side a, B is the angle opposite side b, and C is the angle opposite side c.
Law of cosine can also be written as:
b² = a² + c² − 2ac cos B
a² = b² + c² − 2bc cos A
And law of sine can also be written as:
a / (sin A) = b / (sin B) = c / (sin C)
(Notice that when C = 90°, law of cosine becomes Pythagorean theorem.)
Triangle theorems, like the Pythagorean theorem, are used to establish relationships between the sides of triangles and solve problems. These theorems are used to find unknown sides of triangles when other sides are known. Understanding and applying these theorems can enhance your problem-solving skills in several disciplines.
Explanation:Theorems about triangles, such as the Pythagorean Theorem, can be used to solve various types of mathematical and real-life problems. The Pythagorean Theorem establishes a relationship between the sides of a right-angled triangle. It states that the square of the hypotenuse (side opposite the right angle, labeled 'c') is equal to the sum of the squares of the other two sides (labeled 'a' and 'b'). This relationship is represented by the equation: a² + b² = c².
To use this theorem in solving problems, usually, two sides of a right triangle are known, and the other side is what we need to find out. For example, if the lengths of a and b are known, then c can be found using the formula c = √a² + b². Similarly, if c and one of the other sides are known, the unknown side can be found by rearranging the Pythagorean theorem equation.
Equipped with the understanding of the Pythagorean theorem and other triangle theorems, you can combine various problem-solving strategies to tackle a vast array of problems. This reasoning skill is useful not only in mathematics but also in science disciplines and in everyday life.
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The radius, diameter, or circumference of a circle is given. Find the missing measures. Round to the nearest hundredth if necessary. r = 7.8 mm, d = ? , C = ? Question 1 options: d = 15.6 mm, C = 24.50 mm d = 15.6 mm, C = 49.01 mm d = 3.9 mm, C = 49.01 mm d = 3.9 mm, C = 24.50 mm
Answer:
Second option:
[tex]d = 15.6\ mm,\ C = 49.01\ mm[/tex]
Step-by-step explanation:
We can observe that the radius of the circle is given. This is:
[tex]r = 7.8\ mm[/tex]
And the missing measures are the diameter of the circle and the circumference.
Since the diameter of a circle is twice the radius, we get that this is:
[tex]d=2r\\\\d=2(7.8\ mm)\\\\d=15.6\ mm[/tex]
To find the circumference of the circle, we can use this formula:
[tex]C=2\pi r[/tex]
Where "r" is the radius of the circle.
Substituting the radius into the formula, we get:
[tex]C=2\pi r\\\\C=2\pi (7.8\ mm)\\\\C=49.01\ mm[/tex]
Find the slope of the line that passes through the points (-1, 1) and (2, 7).
Answer:
2 is your slope
Step-by-step explanation:
Find the slope. Use the slope-formula:
m (slope) = (y₂ - y₁)/(x₂ - x₁)
Let:
(x₁ , y₁) = (-1 , 1)
(x₂ , y₂) = (2 , 7)
Plug in the corresponding numbers to the corresponding variables:
m = (7 - 1)/(2 - (-1))
Simplify:
m = (6)/(2 + 1)
m = 6/3
m = 2
2 is your slope (or rise 2, run 1).
~
Answer:
slope = 2
Step-by-step explanation:
Calculate the slope m using the slope formula
m = (y₂ - y₁ ) / (x₂ - x₁ )
with (x₁, y₁ ) = (- 1, 1) and (x₂, y₂ ) = (2, 7)
m = [tex]\frac{7-1}{2+1}[/tex] = [tex]\frac{6}{3}[/tex] = 2
given a right cylinder where h is the height and r is the radius what does the expression 2pirh represent?
A. Base area
B.Area
C.Lateral area
D.Volume
Answer:
C) Lateral Area
Step-by-step explanation:
A) Base Area
The formula for base area of of a cylinder is pir^2.
B) Aread
The formula for surface area or area of a cylinder is 2pirh x 2pir^2 = 2pir (h + r)
C) Lateral Area
The formula for lateral area of a cylinder is 2pirh
D) Volume
The formula for volume of a cylinder is base area x h or pir^2h
!!
Answer:
C. Lateral area.
Step-by-step explanation:
We have been given that a right cylinder where h is the height and r is the radius. We are asked to determine the meaning of our given expression [tex]2\pi rh[/tex].
We know that lateral surface area of cylinder is equal to the circumference of base of cylinder times the height of the cylinder.
We know that base of cylinder is circular and circumference of circle is [tex]2\pi r[/tex], therefore, the expression [tex]2\pi rh[/tex] represents lateral area of cylinder.
If f(x) = 3х – 15, what is f(4)?
[tex]\huge{\boxed{f(4)=\bf{-3}}}[/tex]
In this case, you are replacing all instances of [tex]x[/tex] with [tex]4[/tex]. [tex]f(4)=3(4)-15[/tex]
Multiply. [tex]f(4)=12-15[/tex]
Subtract. [tex]f(4)=-3[/tex]
find the length of arc JM
Answer:
JM ≈ 12.9 miles
Step-by-step explanation:
The length of the arc is calculated as
arc = circumference × fraction of circle
= πd × [tex]\frac{90}{360}[/tex]
JM = π × 16.4 × [tex]\frac{1}{4}[/tex]
= [tex]\frac{16.4\pi }{4}[/tex] ≈ 12.9
The length of the arc JM will be 12.88 miles.
What is the arc length of the sector?Let r is the radius of the sector and θ be the angle subtends by the sector at the center. Then the arc length of the sector of the circle will be
Arc = (θ/360) 2πr
The diameter is 16.4 miles. Then the radius will be
r = d / 2
r = 16.4 / 2
r = 8.2 miles.
And angle (θ) will be 90 degrees.
Then the length of the arc JM will be
Arc = (90/360) 2π x 8.2
Arc = 12.88 miles
More about the arc length of the sector link is given below.
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