A recursive rule for a geometric sequence:
[tex]a_1\\\\a_n=r\cdot a_{n-1}[/tex]
---------------------------------------------------
[tex]a_1=2;\ a_n=\dfrac{4}{9}a_{n-1}\to r=\dfrac{4}{9}[/tex]
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The exciplit rule:
[tex]a_n=a_1r^{n-1}[/tex]
Substitute:
[tex]a_n=2\left(\dfrac{4}{9}\right)^{n-1}=2\cdot\left(\dfrac{4}{9}\right)^n\cdot\left(\dfrac{4}{9}\right)^{-1}=2\cdot\left(\dfrac{4}{9}\right)^n\cdot\dfrac{9}{4}\\\\\boxed{a_n=\dfrac{9}{2}\cdot\left(\dfrac{4}{9}\right)^n}[/tex]
The height, h(t), in feet of an object thrown into the air with an initial upward velocity of 63 feet per second is given by the formula h(t) = -16t2 + 63t, where t is the time in seconds. What is the height, in feet, of the object after 3 seconds?
Answer:
The height of the object after 3 seconds is 45 feets.
Step-by-step explanation:
The height of an object thrown into the air with an initial upward velocity of 63 feet per second is given as
[tex]h(t)=-16t^2+63t[/tex]
Where, h(t) is height of the object in feet and t is the time in seconds.
We have to find the height of the object after 3 seconds. So, substitute t=3.
[tex]h(3)=-16(3)^2+63(3)[/tex]
[tex]h(3)=-16(9)+189[/tex]
[tex]h(3)=-144+189[/tex]
[tex]h(3)=45[/tex]
Therefore height of the object after 3 seconds is 45 feets.
To calculate the height of an object after 3 seconds using the equation h(t) = -16t^2 + 63t, substitute t with 3 and solve. This results in a height of 45 feet.
To find the height after 3 seconds, we substitute t with 3 into the equation, getting h(3) = -16(3)2 + 63(3). Calculating this step-by-step, we first find the square of 3, which is 9, and then multiply it by -16 to get -144. Next, we multiply 63 by 3 to get 189. Adding these two results together, we end up with 45 feet. Therefore, the height of the object after 3 seconds is 45 feet.
What is the fourth term of the expansion of the binomial (2x + 5)5? A. 10x2 B. 5,000x2 C. 1,250x3 D. 2,000x3
Answer:
B would be the answer for this question.
Step-by-step explanation:
Answer: The fourth term is [tex]5000x^2.[/tex]
Step-by-step explanation: We are given to find the fourth term in the expansion of the following binomial :
[tex]B=(2x+5)^5.[/tex]
We know that
the r-th term in the expansion of the binomial [tex](a+x)^n[/tex] is given by
[tex]T_r=^nC_ra^{n-(r-1)}b^{r-1}.[/tex]
For the given term, we have
n = 5 and r = 4.
Therefore, fourth term is given by
[tex]T_4\\\\=^5C_{4-1}(2x)^{5-(4-1)}5^{4-1}\\\\=^5C_3(2x)^25^3\\\\=\dfrac{5!}{3!(5-3)!}\times4x^2\times125\\\\\\=\dfrac{5\times4}{2\times1}\times 500x^2\\\\=5000x^2.[/tex]
Thus, the fourth term is [tex]5000x^2.[/tex]
Please help! Write the slope-intercept form of the equation for the line.
a. y=-8/7x-3/2
b. y=-3/2x+7/8
c. y=-7/8x-3/2
d. y=7/8x-3/2
Answer:
C
Step-by-step explanation:
The slope intercept of a line is y=mx +b where
m is the slope which is calculated as the vertical distance divided by the horizontal distance between two points.b is the y-intercept for value on the y-axis for which the line crosses it.This graph crosses the y-axis (the vertical line) halfway between -1 and -2. This is -3/2. This means only answers a, c, and d are options.
The graph moves up from -3/2 to its next point at (-4,0). We calculate the slope using:
Slope:[tex]m=\frac{y_2-y_1}{x_2-x_1}[/tex]
We substitute [tex]x_1=0\\y_1=-1.5[/tex] and [tex]x_2=-4\\y_2=2[/tex]
[tex]m=\frac{2-(-1.5)}{-4-0}[/tex]
[tex]m=\frac{2+1.5}{-4}=\frac{3.5}{-4} =-0.875[/tex]
This decimal is equivalent to -7/8. This means C is the answer.
This equation represents a line with a slope of [tex]\( -\frac{7}{8} \)[/tex](meaning the line slopes downward from left to right) and a y-intercept of [tex]\( -\frac{3}{2} \)[/tex](where the line crosses the y-axis). The correct answer is option c
The slope-intercept form of the equation of a line is [tex]\( y = mx + b \),[/tex] where m represents the slope of the line, and b represents the y-intercept (where the line crosses the y-axis).
Let's analyze each option:
a.[tex]\( y = -\frac{8}{7}x - \frac{3}{2} \):[/tex]
- Slope [tex]\( m = -\frac{8}{7} \)[/tex]
- y-intercept [tex]\( b = -\frac{3}{2} \)[/tex]
b. [tex]\( y = -\frac{3}{2}x + \frac{7}{8} \):[/tex]
- Slope [tex]\( m = -\frac{3}{2} \)[/tex]
- y-intercept[tex]\( b = \frac{7}{8} \)[/tex]
c.[tex]\( y = -\frac{7}{8}x - \frac{3}{2} \):[/tex]
- Slope [tex]\( m = -\frac{7}{8} \)[/tex]
- y-intercept [tex]\( b = -\frac{3}{2} \)[/tex]
d.[tex]\( y = \frac{7}{8}x - \frac{3}{2} \):[/tex]
- Slope[tex]\( m = \frac{7}{8} \)[/tex]
- y-intercept[tex]\( b = -\frac{3}{2} \)[/tex]
Among the given options, the correct slope-intercept form is option c:
[tex]\[ \boxed{y = -\frac{7}{8}x - \frac{3}{2}} \][/tex]
Jupiter has 11 more than 4 times as many moons has Neptune. Neptune has 14 moons. Let j equal the number of moons Jupiter has.
Final answer:
Jupiter has 67 moons.
Explanation:
To solve this problem, let's define j as the number of moons Jupiter has. According to the given information, Jupiter has 11 more than 4 times as many moons as Neptune, which has 14 moons. So, we can set up an equation: j = 4n + 11, where n is the number of moons Neptune has. Since Neptune has 14 moons, we can substitute that value into the equation: j = 4(14) + 11. Simplifying further, we get j = 56 + 11 = 67. Therefore, Jupiter has 67 moons.
What is an equation of the line, in point-slope form, that passes through the given point and has the given slope?
Answer: [tex]y -3 = \frac{4}{11} (x - 11)[/tex]
Step-by-step explanation:
We can use the point-slope formula to write the equation of a line given a point on the line and the slope of the line:
Slope = m
Given point (x₁,y₁)
Formula = (y-y₁) = m(x-x₁)
Given point: (11,3); slope: 4/11
Answer : [tex]y - 3 = \frac{4}{11} (x - 11)[/tex]
[tex]\textit{\textbf{Spymore}}[/tex]
Ryan will rent a car for the weekend. He can choose one of two plans. The first plan has an initial fee of $59 and costs an additional $0.08 per mile driven. The second plan has an initial fee of $46 and costs an additional $0.10 per mile driven.A) for what amount of driving do the two plans cost the same? B) What is the cost when the two plans cost the same
Help plz 30 points!!
Answer:
the answer is a
Step-by-step explanation:
solve 14^x+1=36
Round to the nearest ten-thousandth.
Answer:
x = 1.3472 to the nearest ten thousandth.
Step-by-step explanation:
14^x + 1 = 36
14^x = 35
Taking logarithms:-
x ln 14 = ln 35
x = ln 35 / ln 14
= 1.3472.
The solution, rounded to the nearest ten-thousandth, is x = 1.5404.
To solve the equation [tex]14^x + 1 = 36[/tex], we first isolate the exponential term by subtracting 1 from both sides of the equation, which gives us [tex]14^x = 35.[/tex]
The next step is to take the logarithm of both sides of the equation.
We could use any base for the logarithm, but it's common to use base 10 or the natural logarithm base (e). In this case, let's use the natural logarithm:
[tex]ln(14^x) = ln(35)[/tex]
We can then use the property of logarithms which allows us to bring the exponent down as a multiplier:
[tex]x * times ln(14) = ln(35)[/tex]
Now, you can solve for x by dividing both sides by ln(14):
[tex]x = ln(35) / ln(14)[/tex]
Using a calculator, we find the quotient and then round to the nearest ten-thousandth:
x = 1.5404
This value is the solution to the original equation.
Shryia read a 480480-page-long book cover to cover in a single session, at a constant rate. After reading for1.51.5 hours, she had 402402 pages left to read.Let P(t)P(t) denote the number of pages to read PP as a function of time tt (measured in hours).Write the function's formula.P(t)=
Answer:
Function formula P(t) = -52t +480
Step-by-step explanation:
here, P(t) denotes the number of pages to read and t represents the time in hour.
Given the statement: Shryia read a 480-page-long book cover to cover in a single session, at a constant rate. After reading for 1.5 hours, she had 402 pages left to read.
⇒Total number of page in a long book = 480
After reading for 1.5 hours, she had 402 pages left to read.
Then,
Total number of page Shryia read in 1.5 hours is:
[tex]480-402 = 78[/tex]
Constant rate at which she is reading her book = [tex]\frac{78}{1.5} = 52[/tex] page per hour
Then, the function formula is given by:
P(t) = -52t + 480 ; where t is in hours.
Check:
P(t) = -52t + 480
P(1.5) = -52(1.5) + 480 = -78 + 480 = 402 True.
The measure of an angle is 78 less than the measure or its complement.What is the measure of the angle
Answer:
84°
Step-by-step explanation:
2 complementary angles sum to 90°
let x be the angle then complement = x - 78, hence
x + x - 78 = 90 ( add 78 to both sides )
2x = 168 ( divide both sides by 2 )
x = 84
hence the angle is 84°
Simplify: (3–a)·2+a =
Answer:
The simplified form of the given expression is 6-a.
Step-by-step explanation:
The given expression is
[tex](3-a)\cdot 2+a[/tex]
According to distributive property.
[tex]a\cdot (b+c)=ab+ac[/tex]
Use distributive property.
[tex]3(2)-a(2)+a[/tex]
[tex]6-2a+a[/tex]
Combine like terms.
[tex]6+(-2a+a)[/tex]
[tex]6-a[/tex]
Therefore the simplified form of the given expression is 6-a.
The expression 1/2bh gives the area of a triangle where b is the base of the triangle and h is the height of the triangle. What is the area of a triangle with the base of 7cm and a height of 4 cm?
[tex]A_{\triangle}=\dfrac{1}{2}bh\\\\\text{We have}\ b=7cm,\ h=4cm\\\\\text{Substitute:}\\\\A_{\triangle}=\dfrac{1}{2}(7)(4)=\dfrac{28}{2}=\boxed{14\ cm^2}[/tex]
Solve for x: 5 over quantity x squared minus 4 plus 2 over x equals 2 over quantity x minus 2.
x = 8
x = –4
x = 8 and x = –4
No Solution
[tex]\text{The domain}\\\\x\neq0\ \wedge\ x\neq-2\ \wedge\ x\neq2[/tex]
[tex]\dfrac{5}{x^2-4}+\dfrac{2}{x}=\dfrac{2}{x-2}\qquad\text{subtract}\ \dfrac{2}{x-2}\ \text{from obth sides}\\\\\dfrac{5}{x^2-2^2}+\dfrac{2}{x}-\dfrac{2}{x-2}=0\\\\\dfrac{5}{(x-2)(x+2)}+\dfrac{2}{x}-\dfrac{2}{x-2}=0\\\\\dfrac{5x}{x(x-2)(x+2)}+\dfrac{2(x-2)(x+2)}{x(x-2)(x+2)}-\dfrac{2x(x+2)}{x(x-2)(x+2)}=0\\\\\dfrac{5x+2(x^2-4)-2x(x+2)}{x(x-2)(x+2)}=0\\\\\dfrac{5x+2x^2-8-2x^2-4x}{x(x-2)(x+2)}=0\\\\\dfrac{x-8}{x(x-2)(x+2)}=0\iff x-8=0\to\boxed{x=8}\in D[/tex]
If two angles are supplementary, which is the sum of their measurements? A. 45o B. 90o C. 120o D. 180o
Two supplementary angle when added together need to equal 180 degrees.
Answer:
Two supplementary angle when added together need to equal 180 degrees.
Step-by-step explanation:
PLEASE HELP ME! What is the equation of a line that passes through the point (6, 1) and is perpendicular to the line whose equation is y=−2x−6?
Enter your answer in the box.
Answer:
y=0.5x-2
Step-by-step explanation:
if it is perpendicular to the line y=-2x-6, then you know that its slope is the negative reciprocal of that line, and it has a different y intercept which you need to solve for using the point given. You solve by plugging in the x and y values from the point and plugging in the slope into the standard equation, and solving for b, the y intercept
y=0.5x+b
1=0.5(6)+b
1=3+b
-2=b
The equation of the line that passes through the point (6, 1) and is perpendicular to the line whose equation is y=−2x−6 is y = 0.5x - 2.
How to find equation of straight line from concept of perpendicular line ?From the classic definition of straight lines, we know that if we have to find an equation of a straight line being perpendicular to another straight line then the slope of the new equation of straight lines becomes negative reciprocal of the slope of given perpendicular line.
Finding the equation of the required straight line -Mathematically, let m1 be the slope of the new straight line and m be the slope of the given perpendicular line, then we have
m1 = -(1/m)
Now, we have given equation y = -2x - 6
Thus slope of the required equation is say (m1) = -(-1/2) = 0.5
Thus the equation formed is y = (m1)x + c [where c is the y-intercept]
∴ y = 0.5x + c
The point given is (6,1) , thus y = 1 and x = 6
Thus the given equation can be formed as
⇒ 1 = 0.5*6 + c
∴ c = 1 - 0.5*6 = -2
The value of y-intercept of the required straight line is -2
The equation of straight line formed is y = 0.5x - 2.
Thus the equation of the line that passes through the point (6, 1) and is perpendicular to the line whose equation is y = − 2x − 6 is y = 0.5x - 2.
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Help me please!!! 30 points! :)
You are going to flip a coin 8 times. The first 3 times you flip the coin you get tails. What is the probability that all the remaining flips will also be tails?
Answer:
So when you flip the coin 8 times and you get tails 3 of the times then you should do 3 plus 5 and you will get 8 as your total again so then the answer is 5 out of 8 total.
Step-by-step explanation:
The probability that all the remaining flips will be tails is 1/32, or approximately 0.03125.
Explanation:The probability of getting tails on each coin flip is 50 percent since a fair coin has two equally likely outcomes: heads or tails.
If the first three coin flips resulted in tails, the remaining five coin flips are independent events. The probability of getting tails on each of the remaining flips is still 50 percent.
The probability of getting all the remaining flips to be tails is calculated by multiplying the probabilities of each individual flip. Since there are five remaining flips, the probability is 0.5 raised to the power of 5, or 1/32 (approximately 0.03125).
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A formula for electrical circuits states that E=P/\I where E represents the force in volts, P represents power in watts, and I represents current in amps. Solve this formula for I. Then find I when the force E = 3.6 volts and the power P = 45 watts.
Answer:
[tex]I=12.5[/tex] amp
Step-by-step explanation:
The formula is,
[tex]E=\dfrac{P}{I}[/tex]
where,
E = Electromotive force in volts,
P = Power in watts,
I = Current in amps.
Given values are,
E = 3.6 volts,
P = 45 watts,
I = ??
Putting the values,
[tex]\Rightarrow 3.6=\dfrac{45}{I}[/tex]
[tex]\Rightarrow I=\dfrac{45}{3.6}=12.5[/tex] amp
Eileen collected 98 empty cans to recycle and Carl 82 cans. They packed a equal number of cans into each of three boxes. How many cans were in each box?
Answer: 60 cans in each box
Step-by-step explanation:
98+82=180
180/3=60
A satellite travels about 2272 miles in 8 minutes about how many miles does a satellite travel in 3 minutes
Answer:
852 miles
Step-by-step explanation:
We presume the speed is constant, so the satellite will travel 3/8 the distance in 3/8 the time.
... d = (3/8)·(2272 miles) = 852 miles
_____
1 minute is 1/8 of 8 minutes, so 3 minutes is 3/8 of 8 minutes.
Drag and drop numbers into the boxes so that the paired values are in a proportional relationship.
x 1 3 _____ 5 8
y 4 12 16 20 _____
4
12
32
8
36
Answer:
4 and 32
Step-by-step explanation:
We are given paired values for two variables x and y and we are to fill in the missing values such that they are in a proportional relationship.
For x, we have the following paired values:
[tex]1, 3, ___, 5, 8[/tex]
So from the given options, 4 fits the best here which is greater than 3 and lesser than 5.
And for y, we have:
[tex]4, 12, 16, 20, ___[/tex]
Here, 32 fits the best from all the options as it is the next (available) number after 20.
River rambler charges $25 per day to rent a kayak. How much will it cost to rent a kayak for 5 days? Write and solve an equation to solve this problem.
Answer:
C=25d
Step-by-step explanation:
We write an equation where C or cost is my output and d or days is my input. I should be able to put in any number of days and find the cost. Let's gather some data:
River Ramble
Day 1 $25(1)=25 cost
Day 2 $25(2)=50 cost
Day 3 $25(3)=75 cost
Day d $25(d)=C.
Our equation is C=25d.
When 4 is subtracted from the square of a number, the result is 3 times the number. Find the negative solution.
Answer:
-1
Step-by-step explanation:
because if you put it in a equation for you get x^2-4=3x therefore you use the quadratic formula and solve the two answers you get are 4 and -1 so since its negative solution it must be -1 must be the square
The negative solution for a quadratic equation, set up to represent the expression 'When 4 is subtracted from the square of a number, the result is 3 times the number', is -1.
Explanation:The subject of this question is a quadratic equation. If we let n be the number, the provided question can be rewritten as:
n² - 4 = 3n
To solve this equation for n, we rearrange terms to get a standard quadratic equation:
n² - 3n - 4 = 0
We can solve this equation using the quadratic formula:
n = [-b ± sqrt(b² - 4ac)] / (2a)
Substituting the values a = 1, b = -3, c = -4 into the formula, we obtain:
n = [3 ± sqrt((-3)² - 4 × 1 × -4)] / (2 × 1)
which simplifies to:
n = [3 ± sqrt(9 + 16)] / 2 = [3 ± sqrt(25)] / 2 = [3 ± 5] / 2
Thus we have two solutions:
n = 8/2 = 4 and n = -2/2 = -1
Since we are looking for the negative solution, the number we are looking for is -1.
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To take a taxi, it costs \$3.00$3.00 plus an additional \$2.00$2.00 per mile traveled. You spent exactly \$20$20 on a taxi, which includes the \$1$1 tip you left. How many miles did you travel?
Answer:
8 miles
Step-by-step explanation:
Given:
To take a taxi initial cost = $ 3.00
Per mile cost = $ 2.00
Total spent = $20
Tip Given = $ 1
To find:
Total miles traveled=?
Solution:
Now the total cost f the ride was $ 20
As we see that initial cost for all the trips would stay the same
Let someone travels x miles
then according to the given
Initial cost + 2 * miles traveled + tip = total cost
Putting from the given values it becomes
3 + 2 * x + 1 = 20
Now we have to find x here which is the total miles traveled
solving the above equation
3+1+2x=20
4+2x=20
2x=20-4
2x=16
dividing both sides by 2
x =8 miles
so total miles travelled by the person is 8 miles
Answer: 8 miles you traveled.
Step-by-step explanation:
Sketch the plane curve represented by the given parametric equations. Then use interval notation to give the relation's domain and range. X = 2t, y = t2 + t + 3
Answer:
Domain: [tex]( -\infty,\infty )[/tex] and Range: [tex][ -1,\infty )[/tex]
Step-by-step explanation:
We have the parametric equations [tex]x= 2t[/tex] and [tex]y=t^{2}+t+3[/tex].
Now, we will find the values of 'x' and 'y' for different values of 't'.
t : -3 -2.5 -2 -1.5 -1 0 1 1.5 2
[tex]x= 2t[/tex] : -6 -5 -4 -3 -2 0 2 3 4
[tex]y=t^{2}+t+3[/tex] : 9 6.75 5 3.75 3 3 5 6.75 9
Now, we can see that these parametric equations represents a parabola.
The general form of the parabola is [tex]y=ax^{2}+bx+c[/tex].
Now, we have the point ( x,y ) = ( 0,3 ). This gives that c = 3.
Also, we have the points ( x,y ) = ( -2,3 ) and ( 2,5 ). Substituting these in the general form gives us,
4a - 2b + 3 = 3 → 4a - 2b = 0 → b = 2a.
4a + 2b + 3 = 5 → 4a + 2b = 2 → 2a + b = 1 → 2a + 2a = 1 ( As, b = 2a ) → 4a = 1 → [tex]a=\frac{1}{4}[/tex].
So, [tex]b=\frac{1}{2}[/tex].
Therefore, the equation of the parabola obtained is [tex]y=\frac{x^{2}}{4}+\frac{x}{2}+3[/tex].
The graph of this function is given below and we can see from the graph that domain contains all real numbers and the range is [tex]y\geq -1[/tex].
Hence, in the interval form we get,
Domain is [tex]( -\infty,\infty )[/tex] and Range is [tex][ -1,\infty )[/tex]
Answer:
Domain:
[tex](-\infty,\infty)[/tex]
Range:
[tex][2.75,\infty)[/tex]
Step-by-step explanation:
we are given parametric equation as
[tex]x=2t[/tex]
[tex]y=t^2+t+3[/tex]
We can change into rectangular equation
we can eliminate t from first equation and plug into second equation
[tex]x=2t[/tex]
[tex]t=\frac{x}{2}[/tex]
now, we can plug that into second equation
[tex]y=(\frac{x}{2})^2+\frac{x}{2}+3[/tex]
now, we can draw graph
Domain:
we know that
domain is all possible values of x for which any function is defined
we can see that our equation is parabolic
and it is defined for all values of x
so, domain will be
[tex](-\infty,\infty)[/tex]
Range:
we know that
range is all possible values of y
we can see that
smallest y-value is 2.75
so, range will be
[tex][2.75,\infty)[/tex]
How do you recognize if a binomial is a difference of perfect squares and how is the pattern used to factor the binomial?
Answer:
A difference of squares has the following form [tex]a^2-b^2[/tex]. Any two perfect squares connected by subtraction can be factored.
It factors to (a+b)(a-b).
Step-by-step explanation:
A binomial is an expression with only terms where at least one is a term with a variable. When we can factor for difference of squares, we can have two variable terms or just one with a constant.
A difference of squares has the following form [tex]a^2-b^2[/tex]. Any two perfect squares connected by subtraction can be factored.
It factors to (a+b)(a-b).
There are 65 students who walk to West Middle School each day. This is 12.5% 0f the total student at the school. How many students attend West Middle School
Order the numbers from least to greatest. A) 1.5, 1.66, 2.4, 3.25, 3.33 B) 1.5, 1.66, 2.4, 3.33, 3.25 C) 1.66, 1.5, 2.4, 3.25, 3.33 D) 3.25, 3.33, 2.4, 1.66, 1.5
What is the answer to: (51 + 11.22 + 35.92)?
Answer:
It would equal 98.14
Step-by-step explanation:
If a right triangle has sides of length a, b and c and if c is the largest, then it is called the "hypotenuse" and its length is the square root of the sum of the squares of the lengths of the shorter sides (a and b). assume that variables a and b have been declared as doubles and that a and bcontain the lengths of the shorter sides of a right triangle: write an expression for the length of the hypotenuse.
Assuming this is for a programming language like c++, then the expression might look like
c = sqrt(a*a + b*b)
or you can use the pow function (short for power function)
c = sqrt( pow(a,2) + pow(b,2) )
writing "pow(a,2)" means "a^2"; similarly for b as well.
the length of hypotenuse can be found using the formula [tex]c = \sqrt{a^2+b^2}[/tex]
The pytharogas theorem states that:
[tex]hypotenuse^2 = perpendicular^2+ base^2[/tex]
The Pythagorean Theorem relates the length of the legs of a right triangle, labeled a and b, with the hypotenuse, labeled c. The relationship is given by:
[tex]a^2 + b^2 = c^2[/tex]
This can be rewritten, solving for c:
[tex]c = \sqrt{a^2+b^2}\\\\[/tex]
Thus, the length of hypotenuse can be found using the formula [tex]c = \sqrt{a^2+b^2}[/tex]