The velocity vector of the particle when it passes through the point (-3,4) is determined as (-4, 8/3) m/s.
What is the particle's position?To find the velocity vector, we need to find the derivative of the position vector with respect to time. Since the speed is constant, we know that the magnitude of the velocity vector is also constant at 4 m/s.
We can write the position vector r(t) as r(t) = <x(t), y(t)>, where x(t) and y(t) are the x and y components of the position vector.
Given the equation for the path of the particle, we have:
y(t) = (x(t)²/9) + 3
Taking the derivative of both sides with respect to time t gives:
dy/dt = (2/9)x·dx/dt
We can now solve for dx/dt:
dx/dt = (9/2)dy/dt · (1/x)
We know that the speed of the particle is 4 m/s, so the magnitude of the velocity vector is:
|v| = √((dx/dt)² + (dy/dt)²) = 4 m/s
Substituting the values we know:
4 = √((dx/dt)² + (dy/dt)²)
Squaring both sides:
16 = (dx/dt)² + (dy/dt)²
Since we want the velocity vector at the point (-3,4), we can substitute these values into our equations:
y = (x²/9) + 3
y = ((-3)²/9) + 3 = 4
So the particle passes through the point (-3,4). Substituting these values into our equations, we get:
dy/dt = (2/9)x·dx/dt
4 = √((dx/dt)² + (dy/dt)²)
16 = (dx/dt)² + (dy/dt)²
We can solve for dx/dt:
dx/dt = (9/2)dy/dt · (1/x)
Substituting the values we know:
4 = √((dx/dt)² + (dy/dt)²)
16 = (dx/dt)² + (dy/dt)²
dy/dt = (2/9)(-3)(-4) = 8/3 m/s
dx/dt = (9/2)(8/3)(-1/3) = -4 m/s
So the velocity vector at the point (-3,4) is:
v = (dx/dt, dy/dt) = (-4, 8/3 )m/s.
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Colby is making a home video consisting of a 5-minute introduction followed by several short skits. Each skit is 6 minutes long. If Colby's video is 143 minutes long, how many skits are in his video?
answer: 23
explanation - 143 - 5 = 138
(total time - the intro)
138 / 6 = 23
(remaining time divided by skit time)
Theorem: For any real number x, x+∣x−5∣≥5 In a proof by cases of the theorem
A. x≤0 B. x≤5 C. x<5 D. x<0
The theorem holds for all real numbers x and the other case is that x < 5.
option C.
What is a real number?A real number is a value that represents a quantity along a continuous number line. Real numbers can be either rational or irrational. Rational numbers are numbers that can be expressed as a ratio of two integers, such as 1/2, 3/4, or -7/8.
For the given question, if one of the cases is that x > 5, we can determine the other case by considering the definition of absolute value:
If x > 5, then ∣x-5∣ = x-5 (since x-5 is positive),
so x + ∣x-5∣ = x + (x-5) = 2x - 5
which is clearly greater than or equal to 5.
If x < 5, then ∣x-5∣ = -(x-5) (since x-5 is negative),
so x + ∣x-5∣ = x - (x-5) = 5,
which is also greater than or equal to 5.
Therefore, the theorem holds for all real numbers x.
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The complete question is below:
Theorem: For any real number x, x+∣x−5∣≥5. In a proof by cases of the theorem, there are two cases. One of the cases is that x > 5.
What is the other case?
A. x≤0
B. x≤5
C. x<5
D. x<0
The ratio of 2 institutions is 7 : 3 and their sum is 630. Find the number
the first institution has 441 and the second institution has 189.
We can use algebra to solve the problem. We can start by assigning variables to represent the two institutions:
Let x be the multiplier of the ratio.
Then, we have:
7x is the first institution.
3x is the second institution.
From the problem, we know that the sum of the two institutions is 630:
7x + 3x = 630
Simplifying the left side, we get:
10x = 630
Dividing both sides by 10, we get:
x = 63
Now that we know the value of x, we can find the number of each institution:
First institution = 7x = 7(63) = 441
Second institution = 3x = 3(63) = 189
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Philip's meal costs $14.50 at a local restaurant. The server did their job very well and Philip would like to a 20% tip. What amount of money should he leave as a tip?
Answer:
$2.90
Step-by-step explanation:
We know
Philip's meal costs $14.50 at a local restaurant. Philip would like to a 20% tip
20% = 0.2
14.50 x 0.2 = $2.90
So, he should leave $2.90 as a tip.
PLS HELP!!!!!
Find area and perimeter.
Answer:
I think it has an area of 4 and a perimeter of 8, i'm not totally sure but I thought I'd try to help
Answer:
area:4
perimeter:8ft
Step-by-step explanation:
when doing the area, you have to MULTIPLY the length by the width. For example, length:5ft width:7ft so to find the area you have to multiply both and get the answer of 35. so for the sum you have is 2 × 2 = 4
for perimeter, you have to ADD all the sides so if you have a square with all sides of 4 you add them all up which = 16. so for your sum, add them all up = 8ft
Hope it helps I'm not that good at explaining :)
An outdoor racetrack is shaped like an ellipse. The equation of the inner lane is x^2/6241 + y^2/1296=1. Each lane is approximately 1 meter apart. Find the length and width of the second lane, as indicated in the figure. Assume that the center is located at (0,0), that the track is a horizontal ellipse, and that one meter is equivalent to one coordinate unit.
ANSWER CHOICES:
A.
length = 79 m
width = 36 m
B.
length = 81 m
width = 38 m
C.
length = 158 m
width = 72 m
D.
length = 160 m
width = 74 m
The length οf the secοnd lane is apprοximately 160 meters, and the width is apprοximately 74 meters. Sο the cοrrect οptiοn is D.
We can start by nοticing that the given equatiοn [tex]x^2/6241 + y^2/1296 = 1[/tex]is the equatiοn οf an ellipse cantered at the οrigin, with a majοr axis οf length 2a = 279 = 158 and a minοr axis οf length 2b = 236 = 72. This is the equatiοn οf the innermοst lane.
Tο find the equatiοn οf the secοnd lane, we need tο add 1 meter tο bοth sides οf the equatiοn, since each lane is apprοximately 1 meter apart. This gives:
[tex]x^2/6241 + y^2/1296 = 2^2[/tex]
Simplifying, we get:
[tex]x^2/6241 + y^2/1296 = 4[/tex]
This is the equatiοn οf an ellipse with the same center as the first ellipse, but with a majοr axis οf length 2(a+1) = 280 = 160 and a minοr axis οf length 2(b+1) = 237 = 74.
Therefοre, the length οf the secοnd lane is apprοximately 160 meters, and the width is apprοximately 74 meters. Sο the cοrrect οptiοn is D.
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8 to 2 but double the equivalent ratio
Answer:
Step-by-step explanation:
8:2 doubled is 16:4
Find the values of p
for which the integral converges. And evaluate the integral for those values of p
.
a
)
∫
1
0
3
1
x
p
d
x
b
)
∫
[infinity]
e
25
x
(
ln
x
)
p
d
x
a) For the integral ∫01^3 1/x^p dx to converge, p > 0. Evaluating the integral for those values of p gives ∫01^3 1/x^p dx = -1/(p-1).
b) For the integral ∫[infinity]^e 25/x^(lnx)^p dx to converge, p > 1. Evaluating the integral for those values of p gives
∫[infinity]^e 25/x^(lnx)^p dx = -25/(p-1).
Convergence and divergence. The improper integral is said to converge if the limit is both real and a limited integer.
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A school supplier sells boxes of A4 paper for £4, and offers a £2 discount on any order paid for in advance. Write the formula the supplier would use for calculating what to charge for any order paid in advance. Use n to represent the number of boxes purchased.
When calculating the price of an order that has been paid in advance, the school supplier will use the following formula:
4n - 2
The ordering cost formula is useful for calculating the economic order quantity and gives businesses insight into how
much of an item is necessary to meet customer demand and fulfill sales goals.
These ordering costs can include shipping fees, unexpected transportation costs, inspection fees and other expenses
necessary to acquire inventory products. Essentially, any money your company spends to place and receive inventory
orders accounts for the purchasing and ordering costs.
Where n represents the number of boxes purchased.
The school supplier sells boxes of A4 paper for £4, and a £2 discount is given on any order paid for in advance.
Formula for calculating what to charge for any order paid in advance:
To calculate the price for an order paid in advance, the formula 4n - 2 should be used.
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how do you calculate volume
Answer:
Height*width*depth= volume
Points A, B, and C are collinear. Point B is the midpoint of AC. Find the length of AB.
AB = 3x
BC = 4x − 6
Answer:
18 unitsStep-by-step explanation:
As per question, point B is the midpoint of AC.
According to definition of midpoint, AB = BC:
3x = 4x - 64x - 3x = 6x = 6The length of AB is:
AB = 3*6 = 18Answer:
AB = 18
Step-by-step explanation:
To find:-
The length of AB .Answer:-
We are here given that points A , B and C are collinear and B is the midpoint of AC . From the data provided in the question we have;
AB = 3x BC = 4x - 6 .Since B is the midpoint of AC, it will divide AC into two equal parts namely AB and BC. And these parts will be equal to one another. ( By Euclid's first axiom, Things which are half of the same thing are equal to one another.)
So we have;
[tex]\sf:\implies AB = BC \\[/tex]
[tex]\sf:\implies 3x = 4x - 6 \\[/tex]
[tex]\sf:\implies 3x -4x = -6 \\[/tex]
[tex]\sf:\implies -x = -6 \\[/tex]
[tex]\sf:\implies\red{ x = 6} \\[/tex]
Finally plug in the value of x, in the given expression of AB as ,
[tex]\sf:\implies AB = 3(x) \\[/tex]
[tex]\sf:\implies AB = 3(6) \\[/tex]
[tex]\sf:\implies \red{AB = 18} \\[/tex]
Hence the value of AB is 18 .
12 ft by 21 ft rectangle. How much carpeting does she need to buy to cover her entire family room?
Answer:
144 feet²
Step-by-step explanation:
12 x 12 = 144
Solve for x. Type your answer as a number
The numerical value of x is 3.
What is the numerical value of x?The Midsegment Theorem states that the segment joining the midpoints of two sides of a triangle is parallel to and half the length of the third side.
This simply means that:
The middle segment = half the third side
From the image:
The middle segment = x + 10The third side = x + 23Using the Midsegment Theorem
x + 10 = 1/2 × (x + 23)
Solve for x
Multiply both sides by 2
2( x + 10 ) = 1/2 ×2 (x + 23)
2( x + 10 ) = x + 23
2x + 20 = x + 23
2x - x = 23 - 20
x = 3
Therefore, the value of x is 3.
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Add. Express your answer in standard form. (Highest exponent first and then descending order)
[tex](2x^{4} -5x^{3} -7) + (-5x^{5} -5x^{2} +9x+17)[/tex]
Answer:
The answer would be -5x^5 + 2x^4 - 5x^3 - 5x^2 + 9x + 10.
What is the area of a sector with a central angle of 90° and a diameter of 20.6 mm?
Use 3.14 for and round your answer to the nearest hundredth.
Enter your answer as a decimal in the box.
mm²
Answer:
37.03 sq. mm.
Step-by-step explanation:
got it right on usa test prep
Name the type of angle indicated
Answer: Acute
Step-by-step explanation:
Acute - Less than 90 degress
Straight - Exactly 180 degrees
Obtuse - More than 90 degrees
Right - Exactly 90 degrees
Mariya was asked to solve StartFraction a over negative 13 EndFraction less-than-or-equal-to negative 16 and then graph the solution. Her work is shown below. In which step, if any, did Mariya make a mistake in her work?
Mariya made a mistake in the third step of her work by multiplying both sides of the inequality by -13 instead of dividing.
What is an inequality?An inequality is a mathematical statement that compares two values or expressions using a symbol such as <, >, ≤, or ≥. It indicates that one value is not equal to another, and may be greater than or less than it.
This resulted in a change of the direction of the inequality from less than or equal to (≤) to greater than or equal to (≥). Therefore, the correct inequality would be a ≥ -13 × -16. The solution of this inequality is a ≥ 208.
When graphing the solution, the solution set will include all values of a greater than or equal to 208, which can be represented by a shaded line to the right of the point 208 on a number line.
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What is -37/76 simplified
Answer:
Already in simplest form. -37/76
Step-by-step explanation:
Nothing changes, it's the most simplified it can be.
answer #
you can't simplify the answer
When you are factoring a polynomiat and the signs of the polynomial are as follows: a x 2
+bx⋅c, the factors will have the following signs: a (t)(t) b (x) (+x) one of each d none of the above Question 2 (1 point) Give one of the prime factors of: x 2
−4x+12 a. (x−4) b (x+4) c) (x−6) d (x+6)
However, the response would be (x - 2 + 2i2)(x - 2 - 2i2) if you are searching for factors with real coefficients.
what is polynomial ?A polynomial is a mathematical expression made up of variables, factors, one or more terms, and non-negative integer exponents used in addition, subtraction, multiplication, and multiplication. Any mathematical object, including numbers, variables, and even functions, can be represented by the variables in a polynomial. In the polynomial equation 3x2 + 2x - 5 as an illustration, the coefficients are 3, 2, and -5, the variable is x, and the exponent is 2. This polynomial has the terms 3x2, 2x, and -5, each of which is denoted by a plus or negative sign. Monomials have one term, binomials have two terms, trinomials have three terms, and so on. Polynomials can be categorized based on how many terms they contain.
given
The solution to the first question is "none of the above," as the signs of the terms in a polynomial are not what determine the signs of the factors.
(x - 2i)(x + 2i) is one of the prime factors of x2 - 4x + 12, where I is the imaginary number.
However, the response would be (x - 2 + 2i2)(x - 2 - 2i2) if you are searching for factors with real coefficients.
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the average bmi (body mass index) value of children 9 years old is 18.4 with a standard deviation of 4.5. if it is known that the distribution is approximately normal, what is the probability that a randomly selected 9-year-old has a bmi greater than 26.8? give your answer to three decimal places.
The probability that a randomly selected 9-year-old has a BMI greater than 26.8 is 0.031. Hence, the answer is 0.031.
What is the formula to calculate standard deviation?
The formula to calculate standard deviation is:
SD = sqrt [(Σ(xi - μ)²) / N]
Where, SD = standard deviation
x = individual data points
μ = mean of the sample
N = sample size
What is the formula for z-score?
The formula for z-score is:
z = (x - μ) / σ
Where,
z = z-score
x = raw score
μ = mean of the population
σ = standard deviation of the population
Given that, The average BMI (body mass index) value of children 9 years old is 18.4 with a standard deviation of 4.5.
To find the probability that a randomly selected 9-year-old has a BMI greater than 26.8, we have to find the z-score.
z = (x - μ) / σz = (26.8 - 18.4) / 4.5z = 1.86
Now, we have to find the probability from the z-table, which is 0.9686. But we have to find the probability greater than 26.8, so we have to subtract it from 1.1 - 0.9686 = 0.0314 ≈ 0.031
Therefore, the probability that a randomly selected 9-year-old has a BMI greater than 26.8 is 0.031. Hence, the answer is 0.031.
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A line has a slope of –12 and passes through the point (6,–9). Write its equation in slope-intercept form.
Answer: y = -12x + 63
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. We are given the slope of the line, which is -12, and a point that the line passes through, which is (6,-9).
Using the point-slope form of a linear equation, we have:
y - (-9) = -12(x - 6)
Simplifying this equation, we get:
y + 9 = -12x + 72
Subtracting 9 from both sides, we get:
y = -12x + 63
Therefore, the equation of the line in slope-intercept form is y = -12x + 63.
70.61% of 736.97cm
Give your answer rounded to 2 DP.
Answer:
520.37 cm
Step-by-step explanation:
apply equation:
x = 736.97(0.7061) = 520.374517
Notice 70.61% is the same as 0.7061 (just divide by 100)
now you need to round to 2 DP
x ≈ 520.37
so the 70.61% of 736.97cm is 520.37 cm
Identify the number as prime, composite or neither. If the number is composite, write it as the product of prime factors 360
Step-by-step explanation:
To determine whether 360 is prime, composite, or neither, we need to check if it has any factors other than 1 and itself.
We can start by finding the prime factorization of 360:
First, we can divide 360 by 2 to get 180:
360 ÷ 2 = 180
Next, we can divide 180 by 2 to get 90:
180 ÷ 2 = 90
We can continue dividing by 2 until we can no longer divide evenly:
90 ÷ 2 = 45
45 ÷ 3 = 15
15 ÷ 3 = 5
5 is a prime number, so we have found the prime factorization of 360:
360 = 2 × 2 × 2 × 3 × 3 × 5
Therefore, we can see that 360 is composite because it has more than two factors. We can write it as the product of its prime factors as shown above.
In summary, the number 360 is composite and its prime factorization is:
360 = 2 × 2 × 2 × 3 × 3 × 5
John and Max work at a sandwich shop. John can make 15 sandwiches per hour, and Max can make 10 sandwiches per hour. Max worked 5 more hours than John and they made a total of 150 sandwiches that day. Determine the number of hours Max worked and the number of hours John worked.
1. Name the 2 variables that are being related in this situation. 2. Write a system of equations to represent the number of hours Max and John worked 3. Solve the system using substitution 4. Interpret the solution of the solution.
20 pts.
Therefore , the solution of the given problem of variable comes out to be John worked for four hours and Max worked for nine.
Variable is what?A variable is a quality that can be evaluated expression and have various values. Height, age, salary, province of birth, academic status, and style of residence range are a few examples of variables.
Here,
Let's use the variables "J" and "M" to denote the respective number of hours done by John and Max. Next, we can formulate the formulae in the following system:
=> J + 5 = M (Max put in five hours more than John did.)
=> 15J + 10M = 150 (They prepared a total of 150 sandwiches) (They made a total of 150 sandwiches)
This system can be resolved using replacement. By deducting 5 from both sides of the first equation, we can find "J" in terms of "M":
=> J = M - 5
Now, we can enter the following formula in place of "J" in the second equation:
=> 15J + 10M = 150
Using J = M - 5 as a substitute, 15(M - 5) Plus 10M = 150.
=> 15M - 75 + 10M Equals 150 (distributing the 15) (distributing the 15)
=> 25M = 225 (adding 75 to both ends) (adding 75 to both sides)
=> M = 9
Max thus put in nine hours of labour. John worked a total of __ hours, so we can re-enter ___ into the first calculation as follows:
=> J + 5 = M
=> J + 5 Equals 9 (M replaced with 9)
=> J = 4
John spent 4 hours at work.
The answer is that John worked for four hours and Max worked for nine.
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At the bakery where Isaac works, one of the cups in a cupcake tray has a radius of 3 centimeters. What is the cup's diameter? centimeters
The diameter οf the cup is 6 centimeters.
What is the diameter οf a circle?A circle's diameter is defined as a line thrοugh the centre that jοins the circumference at bοth ends. It is twice as lοng as the circle's radius. In οther wοrds, the line that splits a circle in half evenly at its centre is the diameter οf the circle.
The diameter οf a circle is equal tο twice its radius.
Therefοre, tο find the diameter οf the cup, we need tο multiply the radius by 2:
Diameter = 2 x Radius
Diameter = 2 x 3 cm = 6 cm
Hence, the diameter οf the cup is 6 centimetres.
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HELP NEEDED QUICK!!
Directions: Calculate the following simple interest problems. Write your answers in the space provided. Use the formula I = P × R × T and round your answers to the nearest cent or the nearest tenth of a percent. Use four decimal places for fractions of time.
(a) I = ?, P = $500, R = 8%, T = 3 months (3/12)
simple interest: $
(b) I = ?, P = $50, R = 12%, T = 1 month (1/12)
simple interest:
cents
(c) I = ?, P = $1,000, R = 18%, T = 24 months (24/12)
simple interest: $
(d) I = ?, P = $600, R = 15%, T = 60 days (60/360)
simple interest: $
(use .1667)
(a) The simplest interest is $12.
(b) The simplest interest is 5 cents.
(c) The simplest interest is $360.
(d) The simplest interest is $15.
What is the interest rate?
In relation to the amount lent, deposited, or borrowed, the amount of interest due each period is expressed as an interest rate. The total interest on a sum lent or borrowed is determined by the principal amount, the interest rate, the frequency of compounding, and the period of time over which it is lent deposited, or borrowed.
The formula of simple interest is I = P × R × T
(a) Given that P = $500, R = 8% = 0.08, T = 3 months = (3/12) years = 1/4 years
The simple interest is
500 × 0.08 × (1/4)
= $12
(b) Given that P = $50, R = 12% = 0.12, T = 1 months = (1/12) years
The simple interest is
50 × 0.12 × (1/12)
=$0.5
= 5 cents
(c) Given that P = $1,000, R = 18% = 0.18, T = 24 months = (24/12) years = 2 years
The simple interest is
1000 × 0.18 × 2
=$360
(d) Given that P = $600, R = 15% = 0.15, T = 60 days = (60/360) years
The simple interest is
600 × 0.15 × (60/360)
=$15
P is principal, R is known as rate of interest.
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A herbalist has 40 oz of herbs costing $4 per ounce. How many ounces of herbs costing $1 per ounce should be mixed with these 40 oz of herbs to produce a mixture costing $2. 20 per ounce?
60 ounces of herbs costing $1 per ounce should be mixed with these 40 oz of herbs to produce a mixture costing $2. 20 per ounce
Let x be the amount of herb that costs $1.00/oz and y be the total amount of the mixture that costs $2.20/oz.
The total weight of the mixture is:
= y=x+40
The total cost of the mixture is:
4(40) + 1x = 2.20y
160 + x = 2.20 (x + 40) .......Substitute y using the expression from the first equation
160 + x = 2.20x + 88 .........Subtract 160 and x from both sides
1.20x = 72 .........Divide both sides by 1.20
x = 60
The herbalist must mix 60 oz of the herb that costs $1.00/oz to the mixture, therefore we can say that:
60 ounces of herbs costing $1 per ounce should be mixed with these 40 oz of herbs to produce a mixture costing $2. 20 per ounce
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a rectangular pizza, 35 cm by 70 cm, is cut into 24 square pieces. two round pizzas, each cut into 12 slices, also give 24 pieces. so that the pizza slices are the same size, what must be the diameter of the round pizzas? round to the nearest tenth. don't include units.
The diameter of each round pizza should be approximately d ≈ 2r ≈ 39.6
Rounding to the nearest tenth, we get a diameter of 39.6 cm.
The rectangular pizza has an area of:
35 cm x 70 cm = 2450 [tex]cm^2[/tex]
Dividing the area by 24 gives us the area of each square piece:
[tex]2450 cm^2 / 24 = 102.08 cm^2[/tex]
For the round pizzas, each slice is a sector of the circle, and the area of each sector is given by:
[tex]A = (θ/360)\pi r^2[/tex]
where θ is the angle of the sector in degrees, and r is the radius (half the diameter) of the circle.
Since each round pizza is cut into 12 slices, each slice covers an angle of:
θ = 360/12 = 30 degrees
We want the area of each slice to be equal to the area of each square piece from the rectangular pizza, which is 102.08 [tex]cm^2[/tex]. Therefore, we can set up the equation:
[tex](30/360)\pi r^2 = 102.08[/tex]
Simplifying, we get:
[tex]\pi r^2/12 = 102.08[/tex]
[tex]\pi r^2 = 1224.96[/tex]
[tex]r^2[/tex] ≈ 391.2
r ≈ 19.8
Therefore, the diameter of each round pizza should be approximately:
d ≈ 2r ≈ 39.6
Rounding to the nearest tenth, we get a diameter of 39.6 cm.
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find each missing length to the nearest tenth
Answer: 4.1
Step-by-step explanation: To find a missing leg you must square both numbers, then subtract the leg you do have from the hypotenuse ( 33.64 - 16.81) to get the squared number for the missing leg ( 16.83). You then have to find the square root of that number and round to the nearest tenth (4.1).
A circular fountain has a radius of 11 feet. Find the area in terms of pi
[tex]\mathcal{A} = \pi r^{2} = \pi (11^{2}) = 121 \pi \ ft^{2}[/tex]
Answer:
121 pi divide to 4
Step-by-step explanation:
the answer is 121 pi divide to 4. please see it on the picture.