The volume of the solid is π/20,
option (A). is correct.
What is volume?Volume is described as a measure of three-dimensional space. It is often quantified numerically using SI derived units or by various imperial or US customary units.
we have that the limits of integration for x are 0 and 1, because the solid lies in the region between x = 0 and x = 1.
Hence, we can say that the volume of the solid is given by:
V = ∫[0,1] (1/4)πx^4 dx
V = (1/4)π ∫[0,1] x^4 dx
V = (1/4)π (1/5) [x^5]0^1
V = (1/20)π
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A baker has small and large bags of sugar for making cakes. The large bag contains 30 cups of sugar and it's 2. 5 times larger than the small bag. The small bag contains enough sugar to make nine cakes and have. 75 cups of sugar remaining
How many cakes can be made with a large bag of sugar?
The number of cakes that can be made with a large bag of sugar, we first need to determine the amount of sugar in a small bag and then calculate the amount of sugar needed for one cake.
1. Find the amount of sugar in a small bag:
Since the large bag contains 30 cups of sugar and is 2.5 times larger than the small bag, we can write the equation:
Small bag = Large bag / 2.5
Small bag = 30 cups / 2.5
Small bag = 12 cups of sugar
2. Determine the amount of sugar needed for one cake:
The small bag contains enough sugar to make 9 cakes and have 0.75 cups of sugar remaining. So, we can subtract the remaining sugar from the total amount in the small bag:
Sugar used for 9 cakes = 12 cups - 0.75 cups
Sugar used for 9 cakes = 11.25 cups
Now, we can find the amount of sugar needed for one cake:
Sugar per cake = Sugar used for 9 cakes / 9
Sugar per cake = 11.25 cups / 9
Sugar per cake = 1.25 cups
3. Calculate the number of cakes that can be made with a large bag of sugar:
Cakes from large bag = Large bag sugar / Sugar per cake
Cakes from large bag = 30 cups / 1.25 cups
Cakes from large bag = 24
Therefore, a baker can make 24 cakes with a large bag of sugar.
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Use even and odd functions to evaluate the following integral. ſ(cosa + 3x4) dx -T
The integral of ſ(cosa + 3x^4) dx simplifies to ∫cos(x) dx, which can be evaluated as sin(x) + C, where C is the constant of integration.To evaluate the integral of ſ(cosa + 3x^4) dx using even and odd functions, we can decompose the integrand into even and odd parts.
Let's first identify the even and odd parts of the integrand. The function cos(x) is an even function because it is symmetric with respect to the y-axis, i.e., cos(-x) = cos(x). On the other hand, the function 3x^4 is an odd function because it is symmetric with respect to the origin, i.e., (-x)^4 = x^4.
We can rewrite the integrand as a sum of even and odd functions:
cos(x) + 3x^4 = (1/2) * (cos(x) + cos(-x)) + (1/2) * (3x^4 - 3(-x)^4)
Now, we can use the properties of even and odd functions to simplify the integral. The integral of an even function over a symmetric interval is equal to twice the integral of the function over half of the interval. Similarly, the integral of an odd function over a symmetric interval is equal to zero.
So, the integral of (1/2) * (cos(x) + cos(-x)) dx is equal to (1/2) * 2 * ∫cos(x) dx, since cos(x) is an even function.
And the integral of (1/2) * (3x^4 - 3(-x)^4) dx is equal to (1/2) * 0, since 3x^4 - 3(-x)^4 is an odd function and the interval of integration is symmetric.
Therefore, the integral of ſ(cosa + 3x^4) dx simplifies to ∫cos(x) dx, which can be evaluated as sin(x) + C, where C is the constant of integration.
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Quadratic function for (1,-3) in vertex form
The quadratic function in vertex form that passes through the point (1, -3) is: f(x) = (x - 1)² - 3
What is vertex form?
Vertex form is a way of expressing a quadratic function of the form:
f(x) = a(x - h)² + k
where (h, k) is the vertex of the parabola, and a is a constant that determines the shape and direction of the parabola.
The quadratic function in vertex form is given by:
f(x) = a(x - h)² + k
where (h, k) is the vertex of the parabola.
We are given the point (1, -3), which lies on the parabola. This means that:
f(1) = -3
Substituting x = 1 into the vertex form of the equation, we get:
f(1) = a(1 - h)² + k
-3 = a(1 - h)² + k
Since we don't know the value of h or a, we can't solve for k directly. However, we can use the vertex form of the equation to find the values of h and k.
The vertex of the parabola is the point (h, k). Since the parabola passes through the point (1, -3), we know that the vertex lies on the axis of symmetry, which is the vertical line x = 1.
Therefore, the x-coordinate of the vertex is h = 1. Substituting this into the equation above, we get:
-3 = a(1 - 1)² + k
-3 = a(0) + k
k = -3
Now that we know the value of k, we can substitute it back into the equation above and solve for a:
-3 = a(1 - h)² + k
-3 = a(1 - 1)² + (-3)
-3 = a(0) - 3
a = 1
Therefore, the quadratic function in vertex form that passes through the point (1, -3) is:
f(x) = (x - 1)² - 3
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A cell phone leans against a wall. The bottom of the phone is 4 inches from the base of the wall, and the top of the phone makes an angle of 52 degrees with the wall. Find the length, x, of the phone so you can buy a new case. Round to the nearest hundreths place
The length of the phone is approximately 6.08 inches, so you can buy a case that fits this size.
To find the length, x, of the phone, we can use trigonometry. We know that the bottom of the phone is 4 inches from the base of the wall, so we can use the tangent function to find the length of the phone.
tangent(52 degrees) = opposite/adjacent
The opposite side is x (the length of the phone) and the adjacent side is 4 inches.
So,
tangent(52 degrees) = x/4
Multiplying both sides by 4, we get:
4 * tangent(52 degrees) = x
Using a calculator, we find that:
x ≈ 6.08 inches
Therefore, the length of the phone is approximately 6.08 inches, so you can buy a case that fits this size.
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Luke has scored a goal in 15 of his 26 soccer games this season and has a hit in 12 of his 16 baseball games this season. Based on the results in his season so far, Luke wants to figure out the probability that he will score a goal in his next soccer game and get a hit in his next baseball game. Enter the probability as a fraction in reduced form
The probability of Luke scoring a goal in his next soccer game and getting a hit in his next baseball game is 45/104 in reduced form.
The probability of Luke scoring a goal in his next soccer game is the ratio of the number of games he scored a goal to the total number of soccer games he played so far. Thus, the probability of scoring a goal in his next game is 15/26.Similarly, the probability of Luke getting a hit in his next baseball game is the ratio of the number of games he had a hit to the total number of baseball games he played so far.
Thus, the probability of getting a hit in his next game is 12/16.Since the events are independent, we can use the product rule to find the probability of both events happening together. Thus, the probability of scoring a goal in his next soccer game and getting a hit in his next baseball game is (15/26) x (12/16) = 45/104 in reduced form.
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Find the value of a2+bc√−d, when
a = –3, b = 2, c = 100, and d = –2
To find the value of a² + bc√(-d) when a = -3, b = 2, c = 100, and d = -2, follow these steps:
Step 1: Substitute the values into the expression.
a² + bc√(-d) = (-3)² + (2)(100)√(-(-2))
Step 2: Simplify the expression.
(-3)² + (2)(100)√(2) = 9 + 200√2
So, the value of a² + bc√(-d) when
a = -3,
b = 2,
c = 100,
d = -2 is 9 + 200√2.
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Mrs. mueller writes an inequality on the board. the table shows the responses of four students for possible values of x.
x>6
student
jacob
kendra
luke
maya
response
6
8
10
12
which student has a correct response to mrs. mueller's inequality?
o jacob
o kendra
o luke
o maya
The inequality given by Mrs. Mueller is x>6, which means that x is greater than 6. To check which student has given the correct response, we need to check if their values of x satisfy the given inequality.
Looking at the table, we see that all four students have given values of x that are greater than 6. However, we need to choose the student who has given the correct response to the inequality.
Jacob has given the response 8, which satisfies the inequality x>6. Kendra has given the response 10, which also satisfies the inequality. Luke has given the response 12, which is also greater than 6 and satisfies the inequality. Maya has given the response 10, which is the same as Kendra's response and also satisfies the inequality.
Therefore, we can say that all four students have given correct responses to Mrs. Mueller's inequality.
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Tina is selling tickets for a fundraiser.
She wants to sell more than $300 worth
of tickets. The inequality 12t> 300 can
be used to determine the number of
tickets, t, she must sell in order to meet
her goal. Which number line represents
the solution to this inequality? (6. 9B |
6. 1A, 6. 1B, 6. 10, 6. 1F)
10
20
30
B
to
10
20
30
+
С
+o
+
10
20
30
D
+
10
O
20
30
The number line that represents the solution to this inequality is 6.10, with an open circle at 25 and shading to the right.
To solve the inequality 12t > 300, we need to isolate t on one side of the inequality. We can do this by dividing both sides by 12:
12t/12 > 300/12
t > 25
This means that Tina must sell more than 25 tickets in order to meet her goal of selling more than $300 worth of tickets.
To represent this solution on a number line, we can start by plotting a point at 25. Since the inequality is greater than (>) and not greater than or equal to (≥), we use an open circle at 25.
Then, we need to shade the area to the right of 25 to represent all the possible values of t that satisfy the inequality. This is because any value of t greater than 25 will make 12t greater than 300.
Out of the answer choices given, the number line that represents the solution to this inequality is 6.10, with an open circle at 25 and shading to the right.
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Mary creates a stack of 10 of piece am and a stack of 8 of piece N. Both stacks have equal volumes. Create an equation relating h and k
Let's assume that the pieces am and N have heights of h_am and h_N respectively, and let k be the number of times the height of piece N fits into the height of piece am (i.e., k is the ratio of the height of piece am to the height of piece N).
We know that the volume of each stack is equal. Let's use the following variables:
- h for the height of each piece of A
- k for the height of each piece of N
- 10 for the number of pieces in stack A
- 8 for the number of pieces in stack N
The equation for the volume of each stack is:
Volume of stack A = h x 10
Volume of stack N = k x 8
Since we know the volumes are equal, we can set the two equations equal to each other:
h x 10 = k x 8
To create an equation relating h and k, we can solve for one variable in terms of the other:
h = (8/10)k
or
k = (10/8)h
Either equation shows how h and k are related to each other. For example, if we know the value of h, we can use the first equation to find k.
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Use the summation formulas to rewrite the expression without the summation notation.
∑ 8i+7/n^2
The expression without the summation notation for ∑ 8i+7/n²2 using the summation formulas is (4n + 3)/2n.
To rewrite the expression without the summation notation, we need to use the summation formulas. We can start by expanding the given summation:
∑ 8i+7/n²2 = 8(1)/n²2 + 8(2)/n²2 + 8(3)/n²2 + ... + 8(n)/n²2 + 7/n²2
Next, we can simplify each term by factoring out 8/n²2:
= (8/n²2)(1 + 2 + 3 + ... + n) + 7/n²2
Using the formula for the sum of the first n positive integers, we have:
= (8/n²2)(n(n+1)/2) + 7/n²2
= (4n² + 4n)/2n² + 7/n²2
= (4n + 3)/2n
Therefore, the expression is (4n + 3)/2n.
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Type the correct answer in the box.
The given equation, V = 1/3 πr²h, solved for h is:
h = 3V / πr²
Subject of formulae: Solving the equation for hFrom the question, we are to solve the give equation for h
From the given information,
The given equation is
V = 1/3 πr²h
To solve the equation for h, we will isolate h
Solving the equation for h
V = 1/3 πr²h
Multiply both sides of the equation by 3
3 × V = 3 × 1/3 πr²h
3V = πr²h
Divide both sides of the equation by πr²
3V / πr² = πr²h / πr²
3V / πr² = h
This can be written as
h = 3V / πr²
Hence, the equation solved for h is:
h = 3V / πr²
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Clayton leased an SUV for his business. The lease cost $421.38 per month for 48 months. He paid a $2,500 deposit, an $85 title fee, and a $235 license fee. Find the total lease cost.
The total lease cost for Clayton's SUV is $23,056.24.
To solve this problemBefore any additional fees or deposits, the total lease cost is $421.38 per month for 48 months, which equals:
Total cost of the lease = $421.38/month x 48 months = $20,236.24
Clayton also paid a $2,500 down payment, a $85 title charge, and a $235 license cost in addition to the monthly lease payments.
The entire cost of the lease is $20,236.24 + $2,500 + $85 + $235 = $23,056.24 in total.
Therefore, the total lease cost for Clayton's SUV is $23,056.24.
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Find all solutions of the equation in the interval [0, 2π). Show formula and steps used, not a calculator problem. (8 csc x - 16)(4 cos x - 4) = 0
The solutions for the equation in the interval [0, 2π) are x = 0, x = π/6, and x = 5π/6.
To find all solutions of the equation (8 csc x - 16)(4 cos x - 4) = 0 in the interval [0, 2π), we can set each factor equal to zero and solve for x separately.
1) 8 csc x - 16 = 0
8 csc x = 16
csc x = 2
Recall that csc x = 1/sin x, so:
1/sin x = 2
sin x = 1/2
In the interval [0, 2π), sin x = 1/2 at x = π/6 and x = 5π/6. So, the solutions for this part are x = π/6 and x = 5π/6.
2) 4 cos x - 4 = 0
4 cos x = 4
cos x = 1
In the interval [0, 2π), cos x = 1 at x = 0 and x = 2π. However, since 2π is not included in the interval, we only have x = 0 as a solution for this part.
Combining both parts, the solutions for the equation in the interval [0, 2π) are x = 0, x = π/6, and x = 5π/6.
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The image shows three sets of stuffed bears. Each set represents a term of the sequence (1, 4, 7,. . . ). An arrangement of stuffed toy bears in groups of 1, 4, and 7
What is the next term in the sequence?
Describe the domain of the sequence. Describe the range of the sequence
The next term in the sequence of the series which have groups of 1, 4, and 7 is 10.
The fundamental concepts in mathematics are series and sequence. A series is the total of all components, but a sequence is an ordered group of items in which repeats of any kind are permitted. One of the typical examples of a series or a sequence is a mathematical progression.
We have the series as 1, 4, 7, ....
First term = a = 1
Common difference = d = 3
Using the formula for the Term is
T = a + (n-1)d
T = 1 + (n-1)3
= 1 + 3n - 3
T = 3n - 2
To find the next term in the series we need to find the 4th term so
T₄ = 3(4) - 2
= 12 - 2
T₄ = 10.
The domain of the sequence T = 3n - 2 is all Real numbers n ∈ Real numbers.
The range is given as
R ∈ (-∞, ∞).
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Answer:
1, 4, and 7 is 10
Step-by-step explanation:
The pattern sequence follows the add 3 rule so, the next term in the sequence will be 10.
The index of the terms of represents the domain of a function, which is { 1, 2, 3, . . .}.
The range includes the terms of the sequence {1, 4, 7, . . .}.
list all the Factors. circle the GCF.
6:
9:
list 7 multiples.Circle the LCM.
5:
2:
Answer:
List all the factors
6: 3, 2, 1 (6)
9:3,(9),1
5:(5),1
2:(2),1
Step-by-step explanation:
What’s the answer I need help pls?
Answer:
(E). y = 2cos(3x)
Step-by-step explanation:
First, amplitude of cos(x) is 1 , then 2cos(x) has amplitude 2
Second, period of cos(x) is 2[tex]\pi[/tex] , then 3 × [tex]\frac{2\pi }{3}[/tex] = 2[tex]\pi[/tex]
So, the answer is y = 2cos(3x)
A scientist recorded the movement of a pendulum for 12 s. The scientist began recording when the pendulum was at its resting position. The pendulum then moved right (positive displacement) and left (negative displacement) several times. The pendulum took 6 s to swing to the right and the left and then return to its resting position. The pendulum’s furthest distance to either side was 7 in. Graph the function that represents the pendulum’s displacement as a function of time. (a) Write an equation to represent the displacement of the pendulum as a function of time. (B) Graph the function. (Please help me answer this for my friend. I am so baffled)
The equation for the displacement of the pendulum as a function of time is: displacement = 7 sin(π/3 t)
How to explain the equationThe motion of a pendulum can be modeled using a sine function:
displacement = A sin(ωt + φ)
where A is the amplitude (the furthest distance from the equilibrium point), ω is the angular frequency (related to the period T by ω = 2π/T), t is time, and φ is the phase angle (determines the starting point of the oscillation).
In this case, the pendulum has an amplitude of 7 inches and a period of 6 seconds (since it takes 6 seconds to swing to one side and then back to the other). Therefore, the angular frequency is:
ω = 2π/T = 2π/6 = π/3
The phase angle is 0, since the pendulum starts at its equilibrium position.
So, the equation for the displacement of the pendulum as a function of time is:
displacement = 7 sin(π/3 t)
where t is measured in seconds and the displacement is measured in inches.
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Please hurry I need it ASAP
Answer: d=2√13
Step-by-step explanation:
You need to use the distance formula or pythagorean. Pythagorean is simpler. Let's use that.
c²=a²+b²
c= distance
a = how far point went in x direction =4
b=how far went in y direction =6
plug in:
d²=4²+6²
d²=16+36
d²=52 take square root of both sides
d=√52
d=√(4*13 4 and 13 are factors of 52
d=2√13 take square root of 4
An oil tank is the shape of a right rectangular prism. The inside of the tank is 36. 5 cm long, 52 cm wide, and 29 cm
high. If 45 liters of oil have been removed from the tank since it was full, what is the current depth of oil left in the
tank?
The current depth of oil left in the tank is approximately 4.64 cm.
The volume of the oil tank can be found by multiplying its length, width, and height:
Volume of the oil tank = length x width x height
= 36.5 cm x 52 cm x 29 cm
= 53,854 cubic cm
If 45 liters of oil have been removed from the tank, the current volume of oil in the tank is:
Current volume of oil = Total volume of tank - Volume of oil removed
= 53,854 cubic cm - 45,000 cubic cm (1 liter = 1000 cubic cm)
= 8,854 cubic cm
Let's assume that the depth of oil left in the tank is x cm. Then the volume of oil left in the tank can be found by multiplying the length, width, and depth of oil:
Volume of oil left in tank = length x width x depth of oil
= 36.5 cm x 52 cm x x cm
= 1906x cubic cm
Now we can set up an equation to find the value of x:
1906x = 8,854
Dividing both sides by 1906, we get:
x = 4.64 cm
Therefore, the current depth of oil left in the tank is approximately 4.64 cm.
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Please asap!!! will give 100 brainlest!!! (there's more than one answer)
select all the correct measures of center and variation for the following data set.
10, 20, 31, 17, 18, 5, 22, 25, 14, 43
a. first quartile = 12
b. iqr = 11
c. median = 19
d. third quartile = 25
e. mad = 7
First quartile is 14, IQR is 14, median is 19, third quartile is 28 and MAD is 7.
a. First quartile = 12 and d. Third quartile = 25 are not necessarily correct measures of quartiles for this dataset. To calculate the quartiles, we need to first order the data set and then find the value(s) that divide it into four equal parts. In this case, the sorted dataset is:
5, 10, 14, 17, 18, 20, 22, 25, 31, 43
The first quartile is the median of the lower half of the data: (5, 10, 14, 17, 18) and is 14.
b. IQR = 11 is not correct. The IQR (Interquartile Range) is the difference between the third quartile and the first quartile, which is 28-14=14 for this dataset.
c. Median = 19 is a correct measure of center.
d. The third quartile is the median of the upper half of the data: (22, 25, 31, 43) and is 28.
e. MAD = 7 is a correct measure of variation.
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What is the solution of |x – 6| ≥ 1? 5 < x < 7 x ≤ –7 or x ≥ –5 x ≤ 5 or x ≥ 7 –7 < x < –5
Answer:
(c) x ≤ 5 or x ≥ 7
Step-by-step explanation:
You want the solution to |x -6| ≥ 1.
UnfoldThe absolute value relation represents two relations, one for the domain x < 6, and one for the domain x ≥ 6.
x < 6In this domain, the inequality becomes ...
-1 ≥ x -6
5 ≥ x . . . . . . add 6
x ≤ 5 . . . . . . . put x on the left
x ≥ 6In this domain, the inequality is ...
x -6 ≥ 1
x ≥ 7
The disjoint solution sets are x ≤ 5 or x ≥ 7.
__
Additional comment
For |x -a| ≤ b, we can "unfold" this to the compound inequality ...
-b ≤ (x -a) ≤ b
copying the inequality symbol to the left side, and writing the opposite of the constant there.
We can do the same thing with the inequality ...
|x -a| ≥ b
but it doesn't really make sense as a compound inequality.
Instead, we have to write it as ...
-b ≥ (x -a) or (x -a) ≥ b
in recognition of the fact that the solution spaces are disjoint.
Anne's Road Paving Company mixed 16 1/4 tons of cement. They used 6 3/4 tons of the cement to pave a street downtown. How much cement did they have left?
Answer is 9.5 tons of cement
Anne's Road Paving Company initially mixed 16 1/4 tons of cement. They used 6 3/4 tons for paving a street downtown. To find the remaining amount of cement, subtract the used amount from the initial amount:
16 1/4 - 6 3/4 = 15 1/4 - 5 3/4 = 9 1/2 tons.
So, they had 9 1/2 tons of cement left.
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7 2 14 3 8 11 5 each time a card is picked it is replaced estimate the expected number of even numbers picked in 35 picks
We can estimate that the expected number of even numbers picked in 35 picks is 15.
To estimate the expected number of even numbers picked in 35 picks, we need to first understand the probability of picking an even number in one pick. Out of the seven given numbers, there are three even numbers (2, 14, 8) and four odd numbers (7, 3, 11, 5). Therefore, the probability of picking an even number in one pick is 3/7.
To find the expected number of even numbers picked in 35 picks, we can multiply the probability of picking an even number in one pick (3/7) by the number of picks (35).
Expected number of even numbers picked = (3/7) x 35 = 15
Therefore, we can estimate that the expected number of even numbers picked in 35 picks is 15. This means that if we were to repeat the process of picking a card and replacing it 35 times, we would expect to pick 15 even numbers on average.
It is important to note that this is an estimate and the actual number of even numbers picked may vary. However, this estimation gives us a good idea of what to expect on average.
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what the size of angle g 82,104,76
Answer:
angle g is 98 degrees.
Step-by-step explanation:
assuming the figure is a quadrilateral,
angle g + 82 + 104 + 76 = 360 ( Property of Quadrilateral)
262 + angle g = 360
angle g = 98 degrees
Emma is making a scale drawing of her farm using the scale 1 centimeter to 2. 5 feet. In the drawing, she drew a well with a diameter of 0. 5 ccentimeter. Which is the closest to the actual circumference of the well?
The circumference of the well is 3.93 ft.
Given, Emma is making a scale drawing of her farm using the scale 1 cm=2.5 ft
Diameter of the well she drew = 0.5 cm
We need to convert the diameter of the well from centimeters to feet, using the given scale.
i.e. 0.5cm = 2.5/2 = 1.25 ft
We know the radius is half of the diameter.
So, r = 1.25/2 = 0.625
We know that the formula for the circumference of a circle is C = 2πr
C = 2*3.14*0.625
= 3.93 ft
Hence, the circumference of the well is 3.93 ft.
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A favorite activity at LNHS is throwing paper
balls into the trashcan while the teacher isn't
looking. Suppose a paper ball is shot from 5 feet
off the ground, and the paper ball reaches a
height of 10 feet after 3 seconds.
*Write the equation that models the height (h)
of the paper ball at any given second (t).
Help me!!
The equation that models the height (h) of the paper ball at any given second (t) is: [tex]h = -16t^2 + 49.67t + 5.[/tex]
To write the equation that models the height (h) of the paper ball at any given second (t), we can use the formula:
[tex]h = -16t^2 + vt + s[/tex]
where v is the initial velocity (in feet per second), s is the initial height (in feet), and t is the time (in seconds).
In this case, we know that the paper ball was shot from 5 feet off the ground, so s = 5. We also know that the paper ball reached a height of 10 feet after 3 seconds, so we can use this information to find the initial velocity:
[tex]h = -16t^2 + vt + s[/tex]
[tex]10 = -16(3)^2 + v(3) + 5[/tex]
10 = -144 + 3v + 5
149 = 3v
v = 49.67 (rounded to two decimal places)
Now we can substitute the values for v and s into the equation:
[tex]h = -16t^2 + vt + s\\h = -16t^2 + 49.67t + 5[/tex]
Therefore, the equation that models the height (h) of the paper ball at any given second (t) is:
[tex]h = -16t^2 + 49.67t + 5.[/tex]
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In QRS, the measure of angle S=90°, the measure of angle Q=6°, and RS = 20 feet. Find the
length of SQ to the nearest tenth of a foot.
R
20
6°
s
Q
X
The length of SQ to the nearest tenth of a foot is approximately 2.1 feet.
To find the length of SQ, we can use trigonometry. First, we can find the measure of angle R by subtracting the measures of angles Q and S from 180°:
R = 180° - 90° - 6° = 84°
Then, we can use the sine function to find the length of SX (which is equal to SQ):
sin(Q) = SQ / RS
sin(6°) = SQ / 20
SQ = 20 * sin(6°)
SQ ≈ 2.07 feet (rounded to the nearest tenth of a foot)
Therefore, the length of SQ to the nearest tenth of a foot is approximately 2.1 feet.
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The slant height if the cone is 13 cm. What is the volume of a cone having a radius of 5 cm and a slant height of 13 cm.
The formula for the volume of a cone is:
V = (1/3)πr^2h
where r is the radius of the base of the cone and h is the height of the cone.
We are given that the radius of the cone is 5 cm and the slant height is 13 cm. We can use the Pythagorean theorem to find the height of the cone:
h^2 = l^2 - r^2
where l is the slant height of the cone. Substituting the given values, we get:
h^2 = 13^2 - 5^2
h^2 = 144
h = 12
Now we can substitute the values of r and h into the formula for the volume of the cone:
V = (1/3)πr^2h
V = (1/3)π(5^2)(12)
V = (1/3)π(25)(12)
V = (1/3)π(300)
V = 100π
Therefore, the volume of the cone is 100π cubic centimeters.
A bag of sweets contains only gobstoppers and sherbert lemons.
There are 3 gobstoppers for every 4 sherbert lemons.
There are 56 sweets in the bag. How many gobstoppers are there?
(1 point) Consider a piece of wire with uniform density. It is the quarter of a circle in the first quadrant. The circle is centered at the origin and has radius 5. Find the centroid (cy) of the wire. =y= (1 point) Compute the total mass of a wire bent in a quarter circle with parametric equations: 2 = 9 cost, y=9 sint, 0
The total mass of the wire is [tex]M = 9\rho * (\pi/2).[/tex]
How to find the total mass of the wire?Using the formula for finding the centroid of a two-dimensional object with uniform density:
cy = (1/Area) * ∫(y*dA)
The equation of the circle is [tex]x^2 + y^2 = 25[/tex]. Solving for y, we get:
[tex]y = \sqrt(25 - x^2)[/tex]
Since the wire is in the first quadrant, the limits of integration are 0 ≤ x ≤ 5 and 0 ≤ y ≤ [tex]\sqrt(25 - x^2).[/tex]
To find the area of the wire, we integrate:
[tex]Area = \int \int dA = \int 0^5 \int 0^{\sqrt(25-x^2)}dy dx[/tex]
[tex]= \int 0^{5 (sqrt(25-x^2))}dx[/tex]
[tex]= (1/2) * [25sin^{(-1)(x/5)} + x\sqrt(25-x^2)] from 0 to 5[/tex]
[tex]= (1/2) * [25\pi/2] = 25\pi/4[/tex]
To find the centroid (cy), we integrate:
[tex]cy = (1/Area) * \int(ydA) = (1/(25\pi/4)) * \int0^5 \int0^{\sqrt(25-x^2)} y dy dx[/tex]
[tex]= (4/25*\pi) * \int0^5 [(1/2)*y^2]_0^{\sqrt(25-x^2)} dx[/tex]
[tex]= (4/25\pi) * \int 0^5 [(1/2)(25-x^2)] dx[/tex]
[tex]= (4/25\pi) * [(25x - (1/3)*x^3)/2]_0^5[/tex]
[tex]= (4/25\pi) * [(255 - (1/3)*5^3)/2][/tex]
[tex]= 50/3[/tex]
Therefore, the centroid of the wire is cy = 50/3.
Now use the formula for the mass of a thin wire for total mass:
M = ∫ρ ds
Since the wire has uniform density, the linear density is constant and can be factored out of the integral:
M = ρ * ∫ds
The differential element of arc length is:
[tex]ds = \sqrt(dx^2 + dy^2) = \sqrt((-9sin t)^2 + (9cos t)^2) dt[/tex]
[tex]= 9\sqrt(sin^2 t + cos^2 t) dt = 9 dt[/tex]
Integrating from 0 to pi/2, we get:
[tex]M = \rho * \int ds = \rho * \int 0^{(\pi/2)} 9 dt[/tex]
[tex]= 9\rho * [t]_0^{(\pi/2)} = 9\rho * (\pi/2)[/tex]
Therefore, the total mass of the wire is [tex]M = 9\rho * (\pi/2).[/tex]
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