The height of the triangle is given as follows:
[tex]h = 2\sqrt{3}[/tex] cm.
The area of the triangle is given as follows:
[tex]A = 4\sqrt{3}[/tex] cm².
What is the Pythagorean Theorem?The Pythagorean Theorem states that in a right-angled triangle, the square of the length of the hypotenuse (the longest side) is equal to the sum of the squares of the lengths of the other two sides.
The theorem is expressed as follows:
c² = a² + b².
In which:
c is the length of the hypotenuse.a and b are the lengths of the other two sides (the legs) of the right-angled triangle.Considering an equilateral triangle, in which all the side lengths are of 4, we have a right triangle in which:
The sides are 2 cm and the height h.The hypotenuse is of 4 cm.Hence the height is obtained as follows:
h² + 2² = 4²
h² = 12
[tex]h = \sqrt{3 \times 4}[/tex]
[tex]h = 2\sqrt{3}[/tex] cm
The area of a triangle is given as half the multiplication of the base and of the height, hence:
[tex]A = 0.5 \times 4 \times 2 \sqrt{3}[/tex]
[tex]A = 4\sqrt{3}[/tex] cm².
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A pitcher contains 13 cups of iced tea. You drink 1. 75 cups of the tea each morning
and 1. 5 cups of the tea each evening. When will you run out of iced tea?
You will run out of iced tea in 5.33 days.
To calculate this, we need to first determine how much tea you drink each day:
1.75 cups in the morning + 1.5 cups in the evening = 3.25 cups per day.
Then, we can divide the total amount of tea by the amount you drink per day to find out how many days the tea will last:
13 cups ÷ 3.25 cups per day ≈ 4 days.
However, we need to account for the fact that you won't run out of tea at the end of the day, so we need to round up to the nearest day:
ceil(4 days) = 5 days.
Finally, we need to account for the partial day on the fifth day, which we can calculate by finding how much tea you drink in the morning before running out:
1.75 cups in the morning - (5 days x 3.25 cups per day) = 0.5 cups.
So, you will run out of iced tea on the fifth day in the evening, after drinking 1.5 cups. Therefore, you will run out of iced tea in 5.33 days.
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Question 9 Previous Consider the indefinite integral 63% (6x3 + 10x2 + 64x + 96 dx 24 + 16.02 Then the integrand has partial fractions decomposition b CC +d + + 22 + 16 where a - 2 a = b = C = du Integrating term by term, we obtain that 16x3 + 10x2 + 64x + 96 dc 24 + 16x2 +C
To do this, we first need to factor the denominator of the integrand into linear factors. In this case, the denominator is given as 24 + 16.02 = 40, which is already a factorization. Therefore, we can write:
∫ (6x3 + 10x2 + 64x + 96)/(24 + 16.02) dx = ∫ [(a/(24 + 16.02)) + (b/(22 + 16))] dx
where a, b are constants that we need to find. To do this, we can use the method of partial fractions, which involves equating the coefficients of like terms on both sides of the equation. Specifically, we can write:
6x3 + 10x2 + 64x + 96 = (a/(24 + 16.02))(22 + 16) + (b/(22 + 16))(24 + 16.02)
Multiplying both sides by the common denominator (24 + 16.02)(22 + 16), we get:
(6x3 + 10x2 + 64x + 96)(24 + 16.02)(22 + 16) = a(22 + 16) + b(24 + 16.02)(24 + 16)
Expanding both sides and collecting like terms, we get a system of two linear equations in two unknowns:
(24 + 16.02)(22 + 16)a + (24 + 16.02)(24 + 16)b = 6(24 + 16.02)(22 + 16) + 10(22 + 16)(24 + 16.02) + 64(24 + 16.02) + 96(22 + 16)
(22 + 16)a + (24 + 16.02)b = 6(22 + 16) + 10(24 + 16.02) + 64 + 96
Solving this system (which involves some algebraic manipulation) gives:
a = -6/5
b = 18/5
Therefore, we can write:
∫ (6x3 + 10x2 + 64x + 96)/(24 + 16.02) dx = (-6/5) ∫ (22 + 16)/(24 + 16.02) dx + (18/5) ∫ (24 + 16.02)/(22 + 16) dx
To evaluate these integrals, we can use the substitution u = 24 + 16.02 in the first integral and u = 22 + 16 in the second integral. This gives:
∫ (6x3 + 10x2 + 64x + 96)/(24 + 16.02) dx = (-6/5) ln|24 + 16.02| + (18/5) ln|22 + 16| + C
where C is the constant of integration. Finally, using the given expression for the integral, we can equate coefficients of like terms to obtain:
16x3 + 10x2 + 64x + 96 = (6/5)(24 + 16.02) ln|24 + 16.02| - (18/5)(22 + 16) ln|22 + 16| + C
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1. A new basketball costs $22. 00 but is on sale for 40% off.
If sales tax is 5%, what is the final cost of the
basketball? Enter your answer as dollars and cents,
The final cost of the basketball, including sales tax, is $15.84.
What is the final price of the basketball, including sales tax, with a 40% discount on the original price of $22.00 and a 5% sales tax rate? Please provide the answer in dollars and cents.The problem states that a new basketball costs $22.00, but it is on sale for 40% off. This means that the customer can purchase the basketball at a discount of 40% from its original price of $22.00.
To calculate the amount of discount, we multiply the original price of the basketball by the discount rate as a decimal:
Discount = 0.40 x $22.00 = $8.80
So, the sale price of the basketball would be:
Sale price = $22.00 - $8.80 = $13.20
Next, the problem states that the sales tax is 5%. Sales tax is a percentage of the sale price of the item, and it is added to the sale price to calculate the final cost of the item.
To calculate the amount of sales tax, we multiply the sale price by the sales tax rate as a decimal:
Sales tax = 0.05 x $13.20 = $0.66
Finally, to calculate the final cost of the basketball, we add the sale price and the sales tax:
Final cost = $13.20 + $0.66 = $15.84
Therefore, the final cost of the basketball, including sales tax, is $15.84.
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The mass of a tumor grows at a rate proportional to its size. The first measurement of its size was 4. 0 grams. Four months later its mass was 6. 76 grams. How large was the tumor six months before the first measurement? If the instrument can detect tumors of mass 1 gram or greater, would the tumor have been detected at that time?
The tumor was approximately 1.47 grams six months before the first measurement. It would not have been detected by the instrument at that time.
Let M(t) be the mass of the tumor at time t (in months) since the first measurement. Since the rate of growth is proportional to the size of the tumor, we have the differential equation dM/dt = kM, where k is the proportionality constant. The general solution to this differential equation is M(t) = Me^(kt), where M is the initial mass of the tumor.
Using the given data, we have M(0) = 4.0 and M(4) = 6.76. Substituting these values into the equation M(t) = Me^(kt), we get the system of equations M = 4.0 and 6.76 = Me^(4k). Solving for M and k, we get M = 4.0 and k = ln(6.76/4.0)/4 ≈ 0.153. Thus, the equation for the mass of the tumor is M(t) = 4.0e^(0.153t).
To find the mass of the tumor six months before the first measurement (i.e., when t = -6), we plug in t = -6 into the equation M(t) = 4.0e^(0.153t) to get M(-6) ≈ 1.47 grams.
Finally, we check whether the tumor would have been detected at that time. Since the mass of the tumor was less than 1 gram at that time, the instrument would not have detected it.
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Miguel draws a square on a coordinate plane. One vertex is located at (5, 4). The length of each side is 3 units. Circle the letter by all the ordered pairs that could be another vertex.
To find the other possible vertices of the square, we need to determine the coordinates of the other three vertices. Since we know that the length of each side is 3 units, we can use this information to determine the distance between the given vertex (5, 4) and the other vertices.
First, we can determine the direction of the square by looking at the given vertex and knowing that the sides of a square are equal in length and perpendicular. Since we know that the side length is 3 units, we can move 3 units to the right to find one possible vertex. This gives us the point (8, 4).
Next, we can move 3 units up to find another possible vertex. This gives us the point (5, 7).
Finally, we can move 3 units to the left to find the last possible vertex. This gives us the point (2, 4).
Therefore, the letter that should be circled by all the ordered pairs that could be another vertex is D, which represents the point (2, 4).
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Sabrina is 4 feet tall and casts a shadow that is 3. 5 feet tall. A nearby pole is 10 feet tall. How tall will the pole’s shadow be if Sabina and the pole are proportional? Leave your answer as a fraction
If Sabina and the pole are proportional, then the pole’s shadow will be 35/4 feet tall.
Since Sabrina is 4 feet tall and casts a 3.5-foot shadow, we can set up a proportion comparing the heights and shadow lengths: Sabrina's height (4 feet) / her shadow (3.5 feet) = pole's height (10 feet) / pole's shadow (x).
This proportion can be represented as: 4/3.5 = 10/x. To solve for x (the length of the pole's shadow), we can cross-multiply:
4 * x = 3.5 * 10
4x = 35
Now, divide both sides by 4:
x = 35/4
So, the pole's shadow will be 35/4 feet long when both Sabrina and the pole are proportional. This fraction represents the pole's shadow length in relation to the given heights and shadow lengths.
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Write 23.4571 correct to
b) the nearest 10
Answer:
20
Step-by-step explanation:
Answer:
23.50
Step-by-step explanation:
23.4571 rounded of to the nearest tenth 23.4+1
Identify the constant of proportionality in the situation
8. A plane travels 462.4 miles in 34 minutes.
Answer:
13.6 miles/minute
Step-by-step explanation:
We Know
A plane travels 462.4 miles in 34 minutes.
Identify the constant of proportionality in the situation..
We Take
462.4 / 34 = 13.6 miles/minute
So, the answer is 13.6 miles/minute
You purchased a home this year for $315,000. You applied for homestead exemption and were able to take off $50,000 of the appraised value for taxes. The taxes in your county are 1.22%. How much do you have to pay in property taxes?
You need to pay $3,243.50 in property taxes.
The appraised value of the house after setting out the homestead exemption is $315,000 - $50,000 = $265,000.
To calculate the assets taxes, we need to multiply the appraised cost by means of the tax rate, which is 1.22% or 0.0122 as a decimal:
property taxes = $265,000 x zero.0122 = $3,243.50
Therefore, you have to pay $3,243.50 in property taxes.
It's far essential to factor in property taxes whilst thinking about the general price of purchasing a home and to recognize the method for applying for exemptions or appealing the appraised cost if necessary.
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I NEED HELP FAST PLEASE!!!
A laptop computer is purchased for $1800. After each year, the resale value decreases by %30. What will the resale value be after 4 years?
Answer:
$432.18
Step-by-step explanation:
First year
$1800
- 540
$1260
Second year
$1260
- 378
$882
Third year
$882
- 264.60
$617.40
Fourth year
$617.40
-185.22
$432.18
i really need help with someone who really understands pythagorean therom please help i will give away brainliest and a lot of points just please help me and can someone please stop reporting this question
Answer:
Perry would need to place his ladder 6.63 ft away from the base of the basketball hoop in order to reach the hoop.
Hope this helps!
Step-by-step explanation:
The Pythagorean theorem is [tex]a^{2} +b^{2} =c^{2}[/tex].
It is given that c is 12 and b is 10, so that would be:
[tex]c^{2} -b^{2} =a^{2}[/tex]
[tex]12^{2} -10^{2} =a^{2}[/tex]
144 - 100 = [tex]a^{2}[/tex]
44 = [tex]a^{2}[/tex]
a = [tex]\sqrt{44}[/tex]
a = 6.6332...
( c is the length of the ladder, b is the height of the hoop, and a is the distance between the ladder and the basketball hoop )
A community center charges x dollars for a summer activity if individuals are signed up before the day of the activity. Individuals who sign up the day of the activity are charged a fee of x plus 0. 20 x dollars. Which expression also represents the fee for signing up the day of the activity, and what does it mean about the fee?.
The expression that represents the fee for signing up the day of the activity is x + 0.20x. This means that fee for signing up the day of the activity is original fee (x) plus an additional 20% of original fee (0.20x).
For example, if the community center charges $50 for signing up before the day of the activity, then the fee for signing up the day of the activity would be:
$50 + ($50 x 0.20) = $50 + $10 = $60
So, individuals who sign up the day of the activity are charged a higher fee than those who sign up before the day of the activity, because they are charged the original fee plus an extra 20% of the original fee. This is often used as an incentive for individuals to sign up early and to help the community center plan and prepare for the activity accordingly.
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Enter the value of c when the expression 21. 2x + c is equivalent of to 5. 3(4x - 2. 6).
The value of c is 21.8.
We can solve for c by simplifying the expression on the right side of the equation and equating it to the expression on the left side.
Starting with the right side, we distribute the 3 across the parentheses:
5 * 3 * (4x - 2.6) = 60x - 13
Now, we can set this equal to the expression on the left and solve for c:
21.2x + c = 60x - 13
c = 60x - 13 - 21.2x
c = 38.8x - 13
However, we are asked to find the value of c, not in terms of x. Since we were not given a specific value for x, we cannot solve for c exactly. Therefore, we can only express the value of c in terms of x, which is c = 38.8x - 13.
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The trip to california from missouri was 2,800 miles one way. how many miles would a round trip take?
The total distance that a round trip will cover is 5,600 miles, under the condition that The trip to California from Missouri was 2,800 miles one way.
After considering all the given data induced by the question we come to the state that here we have to apply the principles of multiplication, to evaluate the distance cover when a round trip is taken from California to Missouri.
Then the calculation is
Round trip distance = 2 × one way distance
Round trip distance = 2 × 2,800 miles
Round trip distance = 5,600 miles
Then the distance covered in the couse of making a round trip from California to Missouri is 5,600 miles.
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Mayumi was asked to determine whether quadrilateral rstu is a trapezoid given the vertices r(-2, 3), s(1, 4), t(1, -4) and u(-2, 1). she noticed that the slopes of ru and st are undefined, so she concluded that the quadrilateral could not be a trapezoid. do you agree? explain.
No, I do not agree with Mayumi's conclusion that the quadrilateral RSTU cannot be a trapezoid just because the slopes of RU and ST are undefined.
RSTU is a trapezoid with RU and ST are parallel and have the same x-coordinates.
A trapezoid is defined as a quadrilateral with at least one pair of parallel sides.
The fact that the slopes of RU and ST are undefined does not necessarily mean that they are not parallel.
As vertical lines have undefined slopes and are parallel to each other.
To determine if RSTU is a trapezoid,
Mayumi should check if any pair of opposite sides are parallel.
The slopes of the two pairs of opposite sides RS and TU, and RU and ST and check if they are equal.
Slope of RS = (4 - 3)/(1 - (-2))
= 1/3
Slope of TU = (1 - (-4))/(-2 - 1)
= -5/3
Slope of RU is undefined (vertical line)
Slope of ST is undefined (vertical line)
Since the slopes of RS and TU are not equal, they are not parallel.
The slopes of RU and ST are undefined does not give us any information about their parallelism.
Check at other properties of the quadrilateral to determine if they are parallel.
One property is the coordinates of the points.
If we draw the quadrilateral, RS and TU are not parallel, but RU and ST are parallel and have the same x-coordinates.
Therefore, quadrilateral RSTU is a trapezoid with bases RS and TU, and legs RU and ST.
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Gas prices are on the rise this summer as more Americans travel, Prices have increased by 40% from last summer's price, which averaged $2. 26 per gallon. Jonah's mom's car takes 14 1/2 gallons of gas to fill up. Find the cost that Jonah's mom would have to spend after the 40% increase in gas prices if she fills up her tank the entire 14 1/2 gallons.
Jonah's mom's would have to spend $45.878 to fill up her tank after the 40% increase in gas prices if her car takes 14 1/2 gallons of gas to fill up.
To find the cost that Jonah's mom would have to spend after the 40% increase in gas prices to fill up her 14 1/2 gallon tank, we need to follow these steps:
1. Find the increased price per gallon by multiplying last summer's price ($2.26) by 1.40 (since there's a 40% increase): $2.26 x 1.40 = $3.164 per gallon.
2. Convert 14 1/2 gallons to a decimal: 14.5 gallons.
3. Multiply the increased price per gallon ($3.164) by the number of gallons needed to fill up the tank (14.5 gallons): $3.164 x 14.5 = $45.878.
So, after the 40% increase in gas prices, Jonah's mom would have to spend $45.878 to fill up her 14 1/2 gallon tank.
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what is the exact volume of a sphere with the radius of 17
Answer: 6,550.6666666666666666666666666667π
Step-by-step explanation:
Hope this helps! :)
A school’s prom committee is composed of 10 students. Seven are girls: Kelly, Karri, Michela, Raquel, Rara, Nadya, and Neesa. Three are boys: Rio, Elke, and Kevin. If they choose a student at random, what is the probability the choose a student whose name begins with "K" P(K name)?
Question 1 options:
30%
70%
40%
50%
The probability of choosing a student whose name begins with "K" is 40%.
To find the probability of choosing a student whose name begins with "K," P(K name), we need to calculate the ratio of students with "K" names to the total number of students in the prom committee.
There are 10 students in the prom committee: 7 girls (Kelly, Karri, Michela, Raquel, Rara, Nadya, Neesa) and 3 boys (Rio, Elke, Kevin). Out of these, 4 students have names that start with "K": Kelly, Karri, Kevin, and Elke (note that Elke has been mistakenly included in the boys' list, but we'll consider it as a "K" name).
To find the probability, we'll use the formula:
P(K name) = (number of students with "K" names) / (total number of students)
P(K name) = 4 / 10 = 0.4 or 40%
So, the probability of choosing a student whose name begins with "K" is 40%.
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You babysat your neighbor's children, and they paid you $45 for 6 hours.
What is the rate?
What is the unit rate?
Write a function rule to represent this situation.
Answer:
The rate is $7.50 per hour.
The unit rate is $7.50 per hour.
This function rule represents the relationship between the number of hours worked (x) and the amount earned (y) at a rate of $7.50 per hour.
Step-by-step explanation:
To find the rate, divide the total amount earned by the number of hours worked. In this case, you earned $45 for 6 hours.
Rate = Total amount earned / Number of hours worked
Rate = $45 / 6 hours
Rate = $7.50 per hour
The rate is $7.50 per hour.
The unit rate refers to the rate per unit of one. In this case, it would be the rate per hour.
Unit Rate = Rate / Number of units
Unit Rate = $7.50 / 1 hour
Unit Rate = $7.50 per hour
The unit rate is $7.50 per hour.
To write a function rule to represent this situation, let's use "x" to represent the number of hours worked and "y" to represent the amount earned:
y = Rate * x
Since the rate is $7.50 per hour, the function rule can be written as:
y = 7.50x
This function rule represents the relationship between the number of hours worked (x) and the amount earned (y) at a rate of $7.50 per hour.
Answer:
Step-by-step explanation:
7.5
Clare made $160 babysitting last summer. She put the money in a saving account that pays 3% interest per year. If Clare doesn't touch the money in her account, she can find the amount she'll have the next year by multiplying her current amount by 1.03.
Write an expression for the amount of money Clare would have after 30 years if she never withdraws money from the account.
Evaluating an exponential function A(x) = 160*(1.03)ˣ we can see that in 30 years she will have:
$388.36
How much money will eh have in 30 years?We know that the initial investment is 160, and the rate per year is 3%.
Then after x years, the value is given by the exponential function.
A(x) = 160*(1.03)ˣ
The amount of money in the account after 30 years is what we get if we evaluate the equation in x = 30, then we will get:
A(30) = 160*(1.03)³⁰
A(30) = 388.36
In 30 years she will have a total amount of 388.36 dollars.
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how long does it take light to travel to earth from sun ? the sun is 9.3 x 10 ^7mi from earth , and the light travels 1.86 x 10^5mi/s
.
It takes approximately 500 seconds for light to travel from the Sun to Earth.
To calculate the time it takes for light to travel from the Sun to Earth, we can use the formula:
time = distance / speed
Given:
Distance from the Sun to Earth = 9.3 x 10^7 miles
Speed of light = 1.86 x 10^5 miles per second
Plugging in the values into the formula, we have:
time = (9.3 x 10^7 miles) / (1.86 x 10^5 miles per second)
To simplify, we can divide the numerator and denominator by 10^5 to cancel out the units:
time = (9.3 x 10^7) / (1.86 x 10^5) seconds
Next, we can divide the numbers in scientific notation:
time = (9.3 / 1.86) x (10^7 / 10^5) seconds
Simplifying further:
time = 5 x 10^2 seconds
Therefore, light takes approximately 500 seconds to travel from the Sun to Earth.
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Heather says that the ratio of bass and violins to cellos is 10 to 5. Allen says the ratio of cellos to bass and violins is 1 to 2. Who is correct?explain your answer
Both ratios provided by Heather and Allen are correct, they are just inverse.
Heather says that the ratio of bass and violins to cellos is 10 to 5. Allen says the ratio of cellos to bass and violins is 1 to 2. To determine who is correct, let's compare the ratios.
1: Simplify Heather's ratio.
Heather's ratio is 10:5, which can be simplified by dividing both sides by 5. This gives a simplified ratio of 2:1 (bass and violins to cellos).
2: Compare the simplified ratios.
Heather's simplified ratio is 2:1, which represents the ratio of bass and violins to cellos. Allen's ratio is 1:2, which represents the ratio of cellos to bass and violins.
3: Analyze the results.
Heather's ratio (2:1) and Allen's ratio (1:2) are inverses of each other. Both ratios are correct, but they represent different perspectives: Heather is expressing the ratio of bass and violins to cellos, while Allen is expressing the ratio of cellos to bass and violins.
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1. An oil spill is spreading such that its area is given by the exponential function A(t) = 250(1. 15)', where A is
the area in square feet and t is the time that has elapsed in days.
(b) By what percent is the oil spill increasing each
t=0?
hour?
The oil spill is increasing by approximately 0.58% per hour.
An oil spill is spreading such that its area is given by the exponential function A(t) = 250(1.15)^t, where A is the area in square feet and t is the time that has elapsed in days. To find the percent increase per hour, first convert the time from days to hours by replacing t with (t/24), since there are 24 hours in a day.
A(t) = 250(1.15)^(t/24)
To determine the hourly percent increase, find the growth factor for one hour (t=1) and subtract 1 to get the percentage:
A(1) = 250(1.15)^(1/24) ≈ 250(1.0058)
The growth factor for one hour is approximately 1.0058. To find the percent increase, subtract 1 and multiply by 100:
(1.0058 - 1) × 100 ≈ 0.58%
The oil spill is increasing by approximately 0.58% per hour.
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4. Question: How can an antiderivative of a velocity function be used to find displacement of the body? Question: If velocity is negative, how does this impact the problem of finding displacement? 5.
If velocity is negative,The body is moving backwards instead of forwards, so the displacement will be measured as a negative value.
An antiderivative of a velocity function can be used to find displacement of the body by integrating the velocity function over a given time interval. The result of this integration will give the total displacement of the body over that time interval.
If the velocity is negative, it means that the body is moving in the opposite direction to the positive direction of the coordinate system. This has an impact on the problem of finding displacement because the displacement will also be negative. In other words, the body is moving backwards instead of forwards, so the displacement will be measured as a negative value.
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1.6.
horseshoe falls is one of the three waterfalls that make up niagara falls near buffalo,
new york. it has an average flow of 7 x 103 cubic meters per second. which value best
represents how much water goes over horseshoe falls hourly?
a. 1.94 x 101 cubic meters per hour
b.
4.20 x 105 cubic meters per hour
c.
2.52 x 10 cubic meters per hour
d.
6.05 x 108 cubic meters per hour
Horseshoe Falls has a flow rate of approximately 2.52 x 10⁷ cubic meters of water per hour, but the given options do not accurately represent this value.
Horseshoe Falls is one of the three waterfalls that make up Niagara Falls near Buffalo, New York. It has an average flow of 7 x 10³ cubic meters per second. To determine the amount of water that goes over Horseshoe Falls hourly, we need to convert the flow rate from cubic meters per second to cubic meters per hour.
There are 3600 seconds in an hour. To convert the flow rate to cubic meters per hour, we simply multiply the flow rate by the number of seconds in an hour:
(7 x 10³ cubic meters/second) x (3600 seconds/hour) = 25.2 x 10⁶ cubic meters/hour
The closest value to this among the given options is:
b. 4.20 x 10⁵ cubic meters per hour
However, there seems to be a typo or error in the provided options. The correct answer should be:
25.2 x 10⁶ cubic meters per hour (approximately 2.52 x 10⁷ cubic meters per hour)
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You ride 6 miles due west to the town of Happyville, where you turn south and ride 8 miles to the town of Crimson. When the sun begins to go down, you decide that it is time to start for home. There is a road that goes directly from Crimson back to Sunshine. If you want to take the shortest route home, do you take this new road, or do you go back the way you came? Justify your decision. How much further would the longer route be than the shorter route? Assume all roads are straight.
It would be more efficient to take the new road from Crimson to Sunshine to get home.
To determine the shortest route home, we can use the Pythagorean theorem, which states that in a right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. In this case, the two sides are the distances you traveled west (6 miles) and south (8 miles).
Using the Pythagorean theorem, we have:
Hypotenuse² = 6² + 8²
Hypotenuse² = 36 + 64
Hypotenuse² = 100
Hypotenuse = √100 = 10 miles
So, the shortest route home (the new road) is 10 miles. If you go back the way you came, you would travel 6 miles west and then 8 miles north, for a total of 14 miles. The longer route (14 miles) is 4 miles longer than the shorter route (10 miles). Therefore, you should take the new road to get home more quickly.
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find the length of the third side if necessary round to the nearest tenth
The third side that we can not see in the image that is shown has a size of 15.
How do you find the hypotenuse of a right triangle when other sides are given?The hypotenuse of a right triangle can be found using the Pythagorean theorem, which states that in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides.
If c^2 = a^2 + b^2
c = √a^2 + b^2
c = √12^2 + 9^2
c = 15
Thus the missing side is 15 from the calculation.
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The change in water level of a lake is shown in the table. How much did the water level change between Weeks 2 and 3?
The rate of change for the water level between Weeks 2 and 3 is: D. 3 7/8.
How to calculate or determine the rate of change or slope of a line?In Mathematics and Geometry, the gradient, rate of change, or slope of any straight line can be determined by using the following mathematical equation;
Rate of change (slope) = (Change in y-axis, Δy)/(Change in x-axis, Δx)
Rate of change (slope) = rise/run
Rate of change (slope) = (y₂ - y₁)/(x₂ - x₁)
By substituting the given data points into the formula for the slope of a line, we have the following;
Rate of change (slope) = (y₂ - y₁)/(x₂ - x₁)
Rate of change (slope) = (1 5/8 + 2 1/4)/(3 - 2)
Rate of change (slope) = (13/8 + 9/4)/(1)
Rate of change (slope) = 3 7/8
Based on the table, the rate of change is the change in y-axis with respect to the x-axis and it is equal to 3 7/8.
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The general form of the primitive of the function f (x) = 6 sin (2) / cos (2) is F (x) = ______ + C
The integral of f(x) can be found by using the substitution u = cos(x), du = -sin(x) dx. Thus, we have:
∫ (6 sin(2x) / cos(2x)) dx = -3 ∫ (2 sin(2x) / cos²(2x)) (-2cos(x)) dx
= 6 ∫ (sin(u) / u²) du
This integral can be evaluated using integration by parts, with u = sin(u) and dv = u⁻² du, giving:
6 ∫ (sin(u) / u²) du = -6 sin(u) / u + 6 ∫ (cos(u) / u) du
Substituting back u = cos(x), we have:
6 ∫ (sin(x) / cos²(x)) dx = -6 tan(x) + 6 ln |cos(x)| + C
Since tan(x) = sin(x) / cos(x) and cos²(x) = 1 - sin²(x), we have:
f(x) = 6 sin(x) / cos²(x) = 6 sin(x) / (1 - sin²(x)) = -6/(sin(x) - 1/sin(x))
Therefore, the general form of the primitive of f(x) is:
F(x) = -6 ln |sin(x) - 1/sin(x)| + C
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1+x Evaluate the repeated integral: 6S6.5. 3 z dz dy dx a) 801 2 729 b) O 8 801 4 d) 729 2 2 729 4 f) None of these.
Without this information, I cannot evaluate the repeated integral or determine the correct answer choice. Please provide additional information so I can assist you better.
To evaluate the repeated integral, we must first understand the given question. It appears that some information is missing, making it difficult to provide a complete answer.
Please provide the complete problem statement, including the limits of integration for each variable (x, y, and z). This will allow me to accurately evaluate the repeated integral and provide you with the correct answer.
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