3 ft-lb of work is needed to stretch it 3 inches beyond the 12ft-lb natural length of the spring.
Here, the integral formula for work and springs should be set up first:
When we look at the integral work for the spring, we know that,
W = [tex]\int\limits^a_b {x} \, dx[/tex] kx dx = k [tex]\int\limits^a_b {x} \, dx[/tex]
we know that,
W = 12,
a = 0,
b = 1
therefore, when we substitute these values in the formula we get,
12 = k [x²/2]¹₀
12 = k (1/2 - 0)
24 = k
now, 3 inches = 1/4th foot = b
so when we substitute this value again with k in the formula we get,
W = [tex]\int\limits^3_0 24 {x} \, dx[/tex]
= 24x²/2
= 12(1/4)²
= 3 ft-lbs
Therefore, we know that when the work required to stretch a spring 1 ft beyond its natural length is 12 ft-lb, 3 ft-lb is needed to stretch it 3 inches beyond its natural length.
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Graph the numbers that are solutions to x−3≤8 and 6x<72 .
can u send a visual picture of the graph?? (94 points)
The graph fοr the sοlutiοns οf the inequalities have been plοtted and attached belοw.
What is an inequality?A relatiοnship in mathematics knοwn as an inequality cοmpares twο numbers οr οther mathematical expressiοns in an unfair manner.
Mοst frequently, size cοmparisοns are dοne between twο numbers οn the number line.
We are given twο inequalities as x - 3 ≤ 8 and 6x < 72.
Nοw, οn sοlving the first inequality, we get⇒ x - 3 ≤ 8⇒ x ≤ 11Similarly, οn sοlving the secοnd inequality, we get⇒ 6x < 72⇒ x < 12
Nοw, οn the graph the sοlutiοns are plοtted and attached belοw. The red pοint represents the secοnd inequality sοlutiοn and the blue represents the first inequality sοlutiοn.
Hence, the graph fοr the sοlutiοns οf the inequalities have been plοtted.
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When a car goes around a curve at twice the speed, the centripetal force on the car doubles. (True or False)
Answer:
True
Step-by-step explanation:
There are 8 finalists in a science fair competition. How many ways can they stand on the stage?
In total, there are 8! (8 factorial) ways that the 8 finalists can stand on the stage. 8! = 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1 = 40,320. This means that there are 40,320 possible combinations in which the 8 finalists can stand on the stage.
To calculate this, we can use the formula n!, which stands for n factorial. n! is the product of all the numbers from n down to 1, so 8! is 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1. This formula can be used to find the total number of combinations that 8 people can stand in on the stage.
It's important to remember that the order matters in this case, so a configuration where the finalists are standing in a line is different from a configuration where they are standing in a circle.
In conclusion, there are 40,320 ways in which the 8 finalists can stand on the stage. This can be determined by using the formula n!, which stands for n factorial.
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through: (-4,-3), parallel to y=2x+4
Whts this in slope int form
Answer:
y=2(x+4)-3
Step-by-step explanation:
To make it parallel to y=2x+4, the slope needs to be the same
Point-slope form is: y=m(x-x1)+y1
In this case,
m = 2
x1 = -4
y1 = -3
So when put into point-slope form, it is y=2(x+4)-3
4. A submarine is traveling a 375 feet below sea leven
rises183 feet and then dives 228 feet if the subm
come safely to the surface at 30 feet per second how many seconds will it take to reach the surface?
97
The submarine will take 14 seconds to safely reach the surface.
What is Speed?Speed is a measure of how quickly an object is moving, calculated as the distance traveled per unit time.
The submarine rises 183 feet, which means its depth decreases by 183 feet. Therefore, its depth is now 375 - 183 = 192 feet below sea level.
The submarine then dives 228 feet, which means its depth increases by 228 feet. Therefore, its depth is now 192 + 228 = 420 feet below sea level.
To reach the surface, the submarine needs to ascend a distance of 420 feet.
The speed of the submarine is given as 30 feet per second. Therefore, the time it will take for the submarine to reach the surface can be calculated as:
[tex]Time = \frac{Distance}{Speed} = \frac{420}{30}[/tex]
= 14 seconds
Therefore, it will take the submarine 14 seconds to safely reach the surface.
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Please help I’ll give brainliest!!
To find the roots of a quadratic equation, ax^2+ bx + c, where a, b, and c are real numbers, Jan uses the quadratic formula. Jan finds that a quadratic equation has 2 distinct roots, but neither are real numbers.
A. Write an inequality using the variables a, b, and c that must always be true for Jan's quadratic equation.
The expression 3+√-4 s a solution of the quadratic equation x^2- 6x +13=0.
B. What is 3+ √-4 written as a complex number?
The inequality will be and [tex]3 + √-4[/tex] will be written as [tex]3 +2i[/tex] as complex number.
What are complex numbers?
In mathematics, a complex number is a number of the form a + bi, where a and b are real numbers and i is the imaginary unit (√-1). Complex numbers are an extension of the real numbers, which include all the numbers that can be represented on a number line.
The real part of a complex number a + bi is the real number a, and the imaginary part is the real number bi.
A. Since Jan found that both roots of the quadratic equation are not real numbers, this means that the discriminant b² - 4ac is negative. Therefore, the inequality that must always be true for Jan's quadratic equation is:
[tex]b² - 4ac < 0[/tex]
B. The expression [tex]3+√-4[/tex] can be written as [tex]3 + 2i,[/tex] where i is the imaginary unit (√-1). This is because √-4 is equal to 2i, so [tex]3+√-4[/tex] can be written as [tex]3 + 2i.[/tex] Therefore, [tex]3+√-4[/tex] written as a complex number is 3 + 2i.
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An insurance company keeps statistics on reported damage to passenger cars. The number of reported injuries for a driver during a year is linked to how long the driver has held a driving licence. The insurance company uses the statistics to set up a probability distribution for two stochastic variables ???? and ???? , for drivers who have held a driving license for up to 3 years.
???? is the number of reported injuries for a driver during a year.
???? is the number of years the driver has held a driving license (???? = 0 means the driver has held a driving license for less than one year).
a)
(i) What is the probability that a random driver has held a license for 2 years and reports 1 injury?
(ii) What is the probability that a random driver has had a driver's license for 1 year and reports 1 injury or has had a driver's license for 2 years and reports 2 injuries?
b) Set up the marginal probability distributions of ???? and ????. What is the probability that a random driver reports 0 injuries? What is the probability that a random driver has had a driver's license for 3 years?
c) If a random driver has not reported any injuries, what is the probability that he has had a driving license for 3 years?
d) Find the expected values ????(????) and ????(????), and the variances ????????????(????) and ????????????(????).
e) Calculate the covariance between ???? and ???? . Calculate the correlation coefficient and give an interpretation of this, related to the task text.
The probability that a random driver reports 0 injuries is 0.25, and the probability that a random driver has had a driver's license for 3 years is 0.25.
a) (i) The probability that a random driver has held a license for 2 years and reports 1 injury is 0.125.
(ii) The probability that a random driver has had a driver's license for 1 year and reports 1 injury or has had a driver's license for 2 years and reports 2 injuries is 0.3125.
b) The marginal probability distributions of ???? and ???? are given in the table below:
????
???? = 0
???? = 1
???? = 2
???? = 3
???? = 0
0.25
0.125
0.0625
0.03125
???? = 1
0.25
0.25
0.125
0.0625
???? = 2
0.25
0.25
0.25
0.125
???? = 3
0.25
0.25
0.25
0.25
The probability that a random driver reports 0 injuries is 0.25, and the probability that a random driver has had a driver's license for 3 years is 0.25.
c) If a random driver has not reported any injuries, the probability that he has had a driving license for 3 years is 0.25.
d) ????(????) = 1, ????(????) = 1.5, ????????????(????) = 0.5, ????????????(????) = 0.9.
e) The covariance between ???? and ???? is 0.225, and the correlation coefficient is 0.45. This shows a positive correlation between the two variables, meaning that an increase in one variable is associated with an increase in the other.
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I dont know how to do this
pls answer if u know with simple working
Answer:
21. Proofs attached to answer
Step-by-step explanation:
Proofs attached to answer
Compound interest. Use the compound interest formula to compute the balance in the following accounts after the stated period of time assuming interest is compounded annually.
$10,000 is invested at an APR of 4% for 10 years
$10,000 is invested at an APR of 2.5% for 20 years
$15,000 is invested at an APR of 3.2% for 25 years
$ 40,000 is invested at an APR of 2.8% for 30 years
compounding more than once a year. Use the appropriate compound interest formula to compute the balance in the following accounts after the stated period of time.
$10,000 is invested for 10 years with an APR of 2% and quarterly compounding
$2000 is invested for 5 years with an APR of 3% and daily compounding
$2000 is invested for 15 years with an APR of 5% and monthly compounding
annual percentage yield (APY) find the annual percentage yield (to the nearest 0.01%) in the following situations.
1. A bank offers an APR of 3.1% compounded daily
2. a bank offers an APR of 3.2% compounded monthly
nick hiked up 10 miles up a hill. he is 8 miles east from his starting point. to the nearest degree, at what angle was the incline of the hill?
The required angle of the incline of the hill, to the nearest degree, is 37 degrees.
How to find the inclined angle?We can use trigonometry to find the angle of the incline of the hill.
we can see that Nick has hiked 10 miles up the hill, and is now at point A. His starting point is at point O, which is 8 miles east of A. Let's call the angle at A, between the incline of the hill and the horizontal ground, theta.
We can use the tangent function to find theta:
[tex]$$\tan(\theta) = \frac{h}{8}$$[/tex]
where h is the height of the hill (the distance from A to the horizontal ground at O).
We know that Nick hiked 10 miles up the hill, so we can use the Pythagorean theorem to find h:
[tex]$$h^2 = 10^2 - 8^2 = 36$$[/tex]
Therefore,
[tex]$$h = \sqrt{36} = 6$$[/tex]
Substituting into the equation for tangent, we get:
[tex]$\tan(\theta) = \frac{6}{8} = 0.75$$[/tex]
To find the angle whose tangent is 0.75, we can use the arctangent function:
[tex]$$\theta = \arctan(0.75)$$[/tex]
Using a calculator, we find that
[tex]$$\theta \approx 36.87^{\circ}$$[/tex]
Therefore, the angle of the incline of the hill, to the nearest degree, is 37 degrees.
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Hot dogs at a carnival cost $2. 99 each plus 7% tax what is the total cost for one hot dog?
What is 600 as a percent out 1900? Please show step by step answers I get confused a lot?
Answer:31.58% i think
Step-by-step explanation:
600 of 1900 can be written as:
600/1900
To find percentage, we need to find an equivalent fraction with denominator 100. Multiply both numerator & denominator by 100
600/1900 x 100/100 = 600x100/1900) x1/100= 31.58/100
Therefore, the answer is 31.58%
United Airlines' flights from Chicago to Seattle are on time 60 % of the time. Suppose 6 flights are randomly selected, and the number on-time flights is recorded.
Round answers to 3 significant figures.
The probability that exactly 3 flights are on time is =
The probability that at most 4 flights are on time is =
The probability that at least 3 flights are on time is =
Therefore , the solution of the given problem of probability comes out to be 3 flights will depart on time is roughly 0.311, the likelihood that at most 4 flights will depart on time is roughly 0.672,
What is probability?Finding the likelihood that a claim is true or that a specific event will occur is the primary objective of the branch of mathematics known as parameter estimation. Any number between range 0 but rather 1, where 1 is usually used to symbolise certainty and 0 is typically used to represent possibility, may be utilized to represent chance. A probability diagram shows the chance that a specific event will occur. .
Here,
=> P(X = k) = (n choose k) * p * k * (1-p) (n-k)
where p is the chance of success and (1-p) is the probability of failure, (n choose k) is the binomial coefficient, and P(X = k) is the probability of k successes in n trials.
The likelihood that precisely 3 planes will depart on time is:
P(X = 3)
=> (6 choose 3) * 0.60 * 0.40 * 0.311
The likelihood that up to 4 planes depart on time is:
=> P(X <= 4) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) + P(X = 4)
= > (6 choose 0) * 0.60^0 * 0.40^6 + (6 choose 1) * 0.60^1 * 0.40^5 + (6 choose 2) * 0.60^2 * 0.40^4 + (6 choose 3) * 0.60^3 * 0.40^3 + (6 choose 4) * 0.60^4 * 0.40^2
≈> 0.672
The likelihood that at least three planes will depart on time is:
P(X >= 3) = 1 - P(X < 3)
= 1 - P(X = 0) - P(X = 1) - P(X = 2)
= 1 - (6 choose 0) * 0.60^0 * 0.40^6 - (6 choose 1) * 0.60^1 * 0.40^5 - (6 choose 2) * 0.60^2 * 0.40^4
≈ 0.695
=> P(X>=3) = 1 - P(X3) = 1 - P(X0) - P(X = 1) - P(X = 2)
=> 1 - (6 choose 0) (6 choose 0) * 0.60^0 * 0.40^6 - (6 choose 1) (6 choose 1) * 0.60 * 0.40 * 6 (select 2) * 0.60 * 0.40 * 4
≈> 0.695
Therefore, the likelihood that precisely 3 flights will depart on time is roughly 0.311, the likelihood that at most 4 flights will depart on time is roughly 0.672, and the likelihood that at least 3 flights will depart on time is roughly 0.695.
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The ratio of dogs to cats is 2:3. If there are 520 dogs, determine how many cats are there?
If the ratio of dogs to cats is 2:3 and there are 520 dogs, the number of cats is 780.
What is the ratio?The ratio refers to the relative size of one quantity or value compared to another quantity or value.
Ratios are proportionate values stated in ratio form using (:), in percentages or fractions.
The ratio of dogs to cats = 2:3
The sum of ratios = 5
The number of dogs based on this ratio = 520 dogs
The total number of dogs and cats based on the above ratio and the number of dogs = 1,300 (520/2 x 5)
The ratio of cats to dogs = 3:2 or 3/5
The number of cats = 780 (1,300 x 3/5)
Thus, using the ratio of dogs to cat, with the number of dogs as 520, there are 780 cats.
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(anyone who answers gets brainliest) Pick the correct trig tool to solve for the variables. There’s a solve for x problem I was struggling with. You don’t have to do them all, thank you
Answer:
(anyone who answers gets brainliest) Pick the correct trig tool to solve for the variables. There’s a solve for x problem I was struggling with. You don’t have to do them all, thank you
Step-by-step explanation:
What is
4y+6x=18
3y-2x=33
Answer:
x = -3 and y = 9
Step-by-step explanation:
Solving system of linear equations by elimination method:4y + 6x = 18 -----------------(I)
3y - 2x = 33 -----------------(II)
Multiply equation (II) by 3.
(I) 4y + 6x = 18
(II)*3 9y - 6x = 99 {Now add the equations}
13y = 117
Divide both sides by 13,
y = 117 ÷ 13
[tex]\boxed{\bf y = 9}[/tex]
Substitute y = 9 in equation (I) and obtain the value of 'x',
4*9 + 6x = 18
36 + 6x = 18
Subtract 36 from both sides,
6x = 18 - 36
6x = -18
Divide both sides by 6
x = -18 ÷ 6
[tex]\boxed{\bf x = -3}[/tex]
[tex] \\ 4y + 6x = 18 \\ \\ 6x= 18 - 4y \\ = x = 3 - \frac{2}{3 }y \\ x = - \frac{2}{3} y+ 3[/tex]
Which correctly describes a cross section of the cube below? Check all that apply.
A cube with 4 centimeter sides.
A cross section parallel to the base is a square measuring 4 cm by 4 cm.
A cross section parallel to the base is a rectangle measuring 4 cm by greater than 4 cm.
A cross section perpendicular to the base through the midpoints of opposite sides is a rectangle measuring 2 cm by 4 cm.
A cross section perpendicular to the base through the midpoints of opposite sides is a square measuring 4 cm by 4 cm.
A cross section that passes through the entire bottom front edge and the entire top back edge is a rectangle measuring 4 cm by greater than 4 cm.
A book sold 42800 copies in its first month of release. Suppose this represents 7. 3 of the number of copies sold to date. How many copies have been sold to date?
In mathematics, the word "of" is also regarded as one of the arithmetic operations, denoting multiplication between brackets. So far, 312440 copies have been sold.
A number or ratio that can be expressed as a fraction of 100 is referred to as a percentage in mathematics. If we need to calculate a percentage of a number, we should divide it by its entirety and then multiply it by 100. The percentage therefore refers to a part per hundred. Per 100 is what the word percent means.
In mathematics, the word "of" is also regarded as one of the arithmetic operations, denoting multiplication between brackets.
We are given that the copies sold on first release date= 42800 copies.
And this is the 7.3 of the number of copies sold on date.
So, let the number of copies sold out on date is x.
Therefore, x= 7.3*42800 OR [tex]x= \frac{73}{10} * 42800= 73*4280= 312440[/tex]
Hence, the number of copies sold out on the date are 312440.
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What is the equation of the line that passes through the point (-9,6) and is perpendicular
to the line 3x-5?
M
Oy--x-3
О У - 3x + 21
Oy - 3x +33
Answer:
y = (-1/3)x + 3.
Step-by-step explanation:
To find the equation of a line perpendicular to a given line, we need to take the negative reciprocal of the slope of the given line.
The slope of the given line Y=3x-5 is 3. Therefore, the slope of a line perpendicular to this line would be -1/3.
Now, using the point-slope form of a line, we can write the equation of the line passing through (-9,6) and having a slope of -1/3:
y - 6 = (-1/3)(x - (-9))
Simplifying:
y - 6 = (-1/3)x - 3
y = (-1/3)x + 3
Therefore, the equation of the line that passes through (-9,6) and is perpendicular to the line Y=3x-5 is y = (-1/3)x + 3.
the perimeter of a rectangular outdoor patio is 106 ft. the length is 9 ft greater than the width. what are the dimensions of the patio?
The perimeter of a rectangular outdoor patio is 106 ft. the length is 9 ft greater than the width. Hence, the dimensions of the patio are 38 ft and 47 ft.
Given that the perimeter of a rectangular outdoor patio is 106 ft. The length is 9 ft greater than the width. Now we need to find the dimensions of the patio.
Step 1: Let's consider the width of the patio be x feet.
The length of the patio is given as 9 ft greater than the width.
So the length of the patio will be (x + 9) feet.
Step 2: The perimeter of a rectangle is given by P = 2(l + w).
So the perimeter of the patio is given as 106 ft.
Thus,2(l + w) = 1062(x + x + 9)
= 1062(2x + 9)
= 1062x + 1821
= 106x = 106 - 182x
= -76 (not possible)
Therefore, x = 38 ftSo the width of the patio is 38 ft The length of the patio is 38 + 9 = 47 ft
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Find the next term of the following sequence.
9, 6, 4, ...
1. 2
2. 8/3
3. 3
Answer:
3
Step-by-step explanation:
You are subtracting 1 less each time, starting at 3:
9 - 3 = 6
6 - 2 = 4
Therefore, you will subtract 1 from 4:
4 - 1 = 3
3 is your answer.
~
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Which statement is true about this quadratic equation?
The correct statement regarding the quadratic equation is given as follows:
C. There are two complex solutions.
How to obtain the number of solutions of the quadratic function?A quadratic equation is modeled by the general equation presented as follows:
y = ax² + bx + c
The discriminant of the quadratic function is given by the equation as follows:
Δ = b² - 4ac.
The numeric value of the coefficient and the number of solutions of the quadratic equation are related as follows:
Δ > 0: two real solutions.Δ = 0: one real solution.Δ < 0: two complex solutions.The coefficients of the function for this problem are given as follows:
a = -2, b = 9, c = -12.
Hence the discriminant is given as follows:
Δ = 9² - 4(-2)(-12)
Δ = -15.
Negative discriminant, hence there are two complex solutions.
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The quadratic equation has two real solutions
Which statement is true about this quadratic equation?Here we have the quadratic equation:
y = -2x^2 + 9x - 12
We want to study the solutions of the equations, so we need to look at the discriminant:
Generally for a*x^2 + b*x + c = 0 the discriminant is b^2 - 4ac
Here it is:
D = 9^2 - 4*-12*-2 = 33
A positive determinant means that there are 2 real solutions so the correct option is A.
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A new cylindrical can with a diameter of 7 cm is being designed by a local company. The surface area of the can is 130 square centimeters. What is the height of the can? Estimate using 3 14 for x, and
round to the nearest hundredth. Apply the formula for surface area of a cylinder SA= 2B+ Ph.
Answer:
see below
Step-by-step explanation:
[tex]SA = 2\pi r \ \ (h+r)[/tex]
[tex]130=2(3.14)(7\div2)(h+(7\div2))[/tex]
[tex]130\div(2(3.14)(7\div2)) = h+(7\div2)[/tex]
[tex]5.91=h+(7\div2)[/tex]
[tex]h = 5.91-3.5[/tex]
Answer Below:
[tex]\bold{x=2.41}[/tex]can yall help me with this I have been up since 7 in the morning and I need some help with this do you think you can help me
Answer: (3/6) then (2/6)
Step-by-step explanation:
FRACTIONS: YOU TIMES IT SO THE DENOMINATOR IS THE SAME APPLY THE NUMBER YOU TIMED BY THE NUMERATOR.
HElp pleaseeeeeeeeeeeee
what are the answer choices?
Answer:
3/4
Step-by-step explanation:
Given the function h(x)=-2√x, which statement is true about h(x)?
O The function is decreasing on the interval (0,0).
O The function is decreasing on the interval (-∞, 0).
O The function is increasing on the interval (0, ∞).
O The function is increasing on the interval (-∞, 0).
Answer:
The statement "The function is decreasing on the interval (0, ∞)" is true about h(x).
To see why, let's take the derivative of h(x) and examine its sign:
h(x) = -2√x
h'(x) = -2/(2√x) = -1/√x
Since √x is always positive, h'(x) is negative for all x > 0. This means that h(x) is decreasing on the interval (0, ∞).
Therefore, the correct answer is: The function is decreasing on the interval (0, ∞).
Use Descartes's Rule of Signs to determine the possible numbers of positive and negative real zeros of the function.
f(x)=2x5−3x2+2x−1
The possible number of positive real zeros of the function is either 2 or 0.f(-x) = −2x5 − 3x2 − 2x − 1The number of sign changes in f(-x) is 1. The possible number of negative real zeros of the function is either 1 or 0.
To determine the possible numbers of positive and negative real zeros of the function f(x) = 2x5 − 3x2 + 2x − 1 using Descartes's Rule of Signs, we should start by writing the polynomial function in descending order of powers of x. After this, we count the number of sign changes in the polynomial function f(x) and find out the possible number of positive real zeros. Similarly, we count the number of sign changes in f(-x) and find out the possible number of negative real zeros .In the given function f(x) = 2x5 − 3x2 + 2x − 1, the polynomial is already in the descending order of powers of x. Therefore,
we count the sign changes in f(x) and f(-x) as follows: f(x) = 2x5 − 3x2 + 2x − 1The number of sign changes in f(x) is 2. Therefore, the possible number of positive real zeros of the function is either 2 or 0.f(-x) = −2x5 − 3x2 − 2x − 1The number of sign changes in f(-x) is 1. Therefore, the possible number of negative real zeros of the function is either 1 or 0. Hence, the possible number of positive real zeros of the function is either 2 or 0 and the possible number of negative real zeros of the function is either 1 or 0.
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Please help!! Thanks
Answer:
[tex]\boxed{\mathtt{Area \approx 104.7m^{2}}}[/tex]
Step-by-step explanation:
[tex]\textsf{We are asked to find the area of 1/3 of a circle.}[/tex]
[tex]\textsf{Let's review the formula needed to find the area of a \underline{whole} circle.}[/tex]
[tex]\large\underline{\textsf{Formula:}}[/tex]
[tex]\mathtt{Area = \pi (radius)^{2}.}[/tex]
[tex]\textsf{We should know that a circle is 360}^{\circ}. \ \textsf{We are given 120}^{\circ} \textsf{of a circle.}[/tex]
[tex]\textsf{The area of \underline{1/3} of a Circle is the area of a whole circle \underline{divided by 3.}}[/tex]
[tex]\textsf{Let's begin solving for the area.}[/tex]
[tex]\large\underline{\textsf{Substitute:}}[/tex]
[tex]\mathtt{Area = \pi (10)^{2}}[/tex]
[tex]\large\underline{\textsf{Evaluate:}}[/tex]
[tex]\mathtt{Area = 100\pi }[/tex]
[tex]\large\underline{\textsf{Divide by 3:}}[/tex]
[tex]\mathtt{\frac{Area}{3} = \frac{100\pi }{3}} [/tex]
[tex]\boxed{\mathtt{Area \approx 104.7m^{2}}}[/tex]
Janelle wants to enlarge a square photograph that she has made so that each side of the new graph will be 1 inch more than twice the original side g. What trinomial represents the area of the enlarged graph? Make sure to explain your answer.
Answer:
Let's start by assigning a variable to the original side length of the square photograph. We'll use g to represent this length.
According to the problem, each side of the new photograph will be 1 inch more than twice the original side length g. This means that the new side length will be:
2g + 1
Since this is a square photograph, all four sides are equal in length. Therefore, the area of the new photograph can be represented by the square of the new side length:
(2g + 1)^2
To simplify this expression, we can use FOIL (First, Outer, Inner, Last) to expand the squared binomial:
(2g + 1)^2 = (2g + 1)(2g + 1)
= 4g^2 + 2g + 2g + 1
= 4g^2 + 4g + 1
So the trinomial that represents the area of the enlarged photograph is:
4g^2 + 4g + 1
To check our work, we can plug in a value for g and compare the areas of the original and enlarged photographs. For example, if g = 2 (meaning the original side length is 2 inches), then the area of the original photograph is:
2^2 = 4 square inches
And the area of the enlarged photograph is:
4(2)^2 + 4(2) + 1 = 25 square inches
This makes sense, since the new photograph has sides of length 2(2) + 1 = 5 inches, and therefore an area of 5^2 = 25 square inches, which is indeed 1 inch more than twice the original area of 4 square inches.