The equation of the parabola with the given dimensions and vertex at the origin is y = -4x^2.
Finding the Equation of a Parabola
To find the equation of a parabola with a vertex at the origin and given dimensions, we can use the standard form of a parabolic equation, which is y = ax^2. In this case, since the parabola opens downward and the vertex is at the origin (0,0), the equation will have the form y = -ax^2. The value of 'a' can be determined using the dimensions provided for the parabola, which are a width of 42 feet (meaning that the points (21,0) and (-21,0) are on the parabola) and a height of 84 feet (the y-coordinate at the vertex).
Since the point (21, 0) lies on the parabola, substituting it into the equation y=-ax^2 gives us 0 = -a(21)^2, which leads us to find that a = -84/(21)^2. Substituting the value of 'a' back into the equation gives us the final equation of the parabola: y = -84/(21)^2
x^2.
The equation of the parabola is: [tex]\[ y = -\frac{2}{21}x^2 + 42 \][/tex] .
To find the equation of the parabola with its vertex at the origin, we can use the standard form of a parabolic equation, which is [tex]\( y = ax^2 \).[/tex]
Given that the parabola opens downwards, we know that a must be negative. To determine the value of ( a ), we need to find a point on the parabola.
We're given the dimensions of the arch: 84 feet high and 42 feet wide at the base. Since the arch is symmetrical, the highest point is at the midpoint of the base, which is ( x = 0 ). At this point,( y = 84 ).
So, substituting the coordinates of this point into the equation, we get:
[tex]\[ 84 = a \times 0^2 \][/tex]
This simplifies to ( 84 = 0 ), which doesn't give us any useful information. Instead, we need to consider another point on the parabola.
Since the arch is symmetric, we can choose a point where ( x = 21 ) (half of the width of the base), and ( y = 0 ).
Substituting these coordinates into the equation, we get:
[tex]\[ 0 = a \times (21)^2 \][/tex]
0 = 441a
Dividing both sides by 441, we find ( a = 0 ). However, this seems incorrect, as it would mean the arch is just a straight line, which it isn't. This suggests that our choice of coordinates may not be correct.
Let's reconsider. The midpoint of the base is x = 0, but the highest point might not be there. Instead, let's choose a point where ( x = 0 ) and ( y = 42 ), as this is the highest point of the arch.
Substituting these coordinates into the equation, we get:
[tex]\[ 42 = a \times 0^2 \][/tex]
42 = 0
This also doesn't give us useful information. It seems we might have approached this problem incorrectly. Let's try a different strategy.
Since we know the arch is a parabolic shape, and the parabola opens downwards, we can write its equation in the form:
[tex]\[ y = ax^2 + c \][/tex]
To find the values of a and c , we need two points on the parabola. We already have one: the highest point of the arch, which is at x = 0 and (y = 42 ).
Now, we need to find another point. Since the arch is symmetric, we can use any point along the base. Let's choose the point where x = 21 , which is half of the width of the base. At this point, y = 0 .
Substituting these points into the equation, we get:
[tex]\[ 42 = a \times 0^2 + c \][/tex]
[tex]\[ 0 = a \times 21^2 + c \][/tex]
The first equation simplifies to ( c = 42 ).
Substituting this value of ( c ) into the second equation, we get:
[tex]\[ 0 = a \times 21^2 + 42 \][/tex]
Solving for a :
[tex]\[ a \times 441 = -42 \][/tex]
[tex]\[ a = \frac{-42}{441} \][/tex]
[tex]\[ a = -\frac{2}{21} \][/tex]
So, the equation of the parabola is:
[tex]\[ y = -\frac{2}{21}x^2 + 42 \][/tex]
The best leaper in the animal kingdom is the puma, which can jump to a height of 12 ft when leaving the ground at an angle of 45 degrees. with what speed in si units, must the animal leave the ground to reach that height?
On average, Donna's Cafe has 84 customers, which represents 24% of the total approved occupancy by the fire department. a. What is the total approved occupancy of the cafe by the fire department?
A parallelogram is cut out of a 12-inch by 8-inch sheet of paper. There are four right triangle remnants. Two have the dimensions 2 inches by 9 inches, and the other two have the dimensions 3 inches by 6 inches. The resulting parallelogram has a base of approximately 9.22 inches. Complete the following steps to calculate the altitude of the parallelogram using area methods. The area of the sheet of paper is square inches. The combined area of the triangle cutouts is square inches. The area of the parallelogram is square inches. The altitude of the parallelogram rounded to two decimals is square inches.
Answer:
96
36
60
6.51
Step-by-step explanation:
The area of a shape is the amount of space it occupies.
The area of the paper is 96 square inchesThe combined area of the triangle cutouts is 36 square inchesThe area of the parallelogram is 60 square inchesThe altitude of the parallelogram is 6.51 inchesThe dimension of the paper is given as: 12-inch by 8-inch
So, its area is:
[tex]\mathbf{A_1 =12 \times 8}[/tex]
[tex]\mathbf{A_1 =96}[/tex]
The dimensions of the 4 right triangles are: Two 2 inches by 9 inches, and two 3 inches by 6 inches
So, the combined area is:
[tex]\mathbf{A_2 = 2 \times \frac 12 \times 2 \times 9 + 2 \times \frac 12 \times 3 \times 6}[/tex]
[tex]\mathbf{A_2 = 36}[/tex]
The area of the parallelogram is the difference between the areas of the paper and the four right triangles.
So, we have:
[tex]\mathbf{A_3 = A_1 - A_2}[/tex]
[tex]\mathbf{A_3 = 96 - 36}[/tex]
[tex]\mathbf{A_3 = 60}[/tex]
The area of a parallelogram is:
[tex]\mathbf{Area = Base \times Altitude}[/tex]
The base is given as 9.22
So, we have:
[tex]\mathbf{60 = 9.22\times Altitude}[/tex]
Divide both sides by 9.22
[tex]\mathbf{6.51 = Altitude}[/tex]
Rewrite as:
[tex]\mathbf{Altitude = 6.51}[/tex]
Hence, the altitude of the parallelogram is 6.51 inches
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Solve the equation for a. 55a – 44a = 11
Help with this math question
the directions for mixing water and powdered milk say that one should use 3 cups of water for each cup of powdered milk. how many cups of powdered milk are needed if 5 cups of water are used
The sum of 5 consecutive even number is 100 the middle number is
100/5 =20
16 + 18 +20 +22 +24 = 100
middle number is 20
One 16-ounce bottle of an energy drink has an average of 500 mg of caffeine with a standard deviation of 25 mg. In a carton containing 30 bottles, what is the standard deviation of the average amount of caffeine?
Final answer:
The standard deviation of the average amount of caffeine in a carton containing 30 bottles of an energy drink is approximately 4.564 mg.
Explanation:
To calculate the standard deviation of the average amount of caffeine in a carton containing 30 bottles of an energy drink, we can use the concept of the sampling distribution of the sample mean.
When independent random samples of a fixed size are taken from a population, the standard deviation of the sample means (σx-bar) is equal to the population standard deviation (σ) divided by the square root of the sample size (n), which is also known as the standard error of the mean.
The formula to find the standard deviation of the sample means is:
σx-bar = σ / √n
In this case, the population standard deviation is given as 25 mg of caffeine, and the sample size is 30 bottles.
So, we calculate the standard deviation of the average amount of caffeine as follows:
σx-bar = 25 mg / √30
σx-bar ≈ 25 mg / 5.477
σx-bar ≈ 4.564 mg (rounded to three decimal places)
Thus, the standard deviation of the average amount of caffeine in a carton containing 30 bottles is approximately 4.564 mg.
You buy a pair of jeans at a department store. Jeans 39.99 Discount -10.00 Subtotal 29.99 Sales tax 1.95 Total 31.94 a. What is the percent of discount to the nearest percent? The percent of discount is ___%. b. What is the percent of sales tax to the nearest tenth of a percent? The percent of sales tax is ___% c. The price of the jeans includes a 60% markup. After the discount, what is the percent of markup to the nearest percent? The percent of markup is ___%
The discount offered on the jeans is 25 percent. The sales tax charged on the jeans is 6.5 percent. Markup on the jeans after allowing discount is 60 percent.
a. To find the percent of discount, we need to calculate the actual discount and divide it by the original price, then multiply by 100. From the given information, the original price of the jeans is $39.99 and the discount is $10.00. Therefore, the actual discount is $10.00 and the original price is $39.99. The percent of discount to the nearest percent is:
Percent of discount = (actual discount / original price) x 100
Percent of discount = (10.00 / 39.99) x 100
Percent of discount = 25%
Therefore, the percent of discount to the nearest percent is 25%.
b. To find the percent of sales tax, we need to divide the sales tax by the subtotal, then multiply by 100. From the given information, the subtotal is $29.99 and the sales tax is $1.95. Therefore, the percent of sales tax to the nearest tenth of a percent is:
Percent of sales tax = (sales tax / subtotal) x 100
Percent of sales tax = (1.95 / 29.99) x 100
Percent of sales tax = 6.5%
Therefore, the percent of sales tax to the nearest tenth of a percent is 6.5%.
c. The price of the jeans includes a 60% markup, which means that the selling price is 160% of the cost price. Let x be the cost price of the jeans. Then, the selling price of the jeans is 1.6x. After the discount of $10.00, the selling price becomes 1.6x - 10.00. The percent of markup after the discount to the nearest percent is:
Percent of markup = ((selling price - cost price) / cost price) x 100
Percent of markup = (((1.6x - 10.00) - x) / x) x 100
Percent of markup = (0.6x - 10.00) / x x 100
Percent of markup = 60 - (1000 / x)
Percent of markup = 60%.
So, the percent of markup after the discount is approximately 60%.
Jessica and Franklin are volunteering to paint one of their community member’s fences for service hours. If Jessica could complete the job in 6 hours and Franklin could complete the job in 4.5 hours, how long would it take them to complete the job together, to the nearest minute?
We deal from a well-shuffled 52-card deck. calculate the probability that the 13th card is the first king to be dealt.
Final answer:
The probability that the 13th card is the first king to be dealt is 1/52.
Explanation:
To calculate the probability that the 13th card is the first king to be dealt from a well-shuffled 52-card deck, we need to consider the number of favorable outcomes and the total number of possible outcomes. There are 4 kings in the deck, so the probability of the first king being the 13th card is 1/52.
Final answer:
The probability that the 13th card is the first king to be dealt in a shuffled 52-card deck is 1/13.
Explanation:
In a well-shuffled 52-card deck, the probability that the 13th card is the first king to be dealt can be calculated as follows:
First, let's determine the number of favorable outcomes. Since there are 4 kings in the deck, there are 4 possible kings that can be the first king dealt.
Next, let's determine the number of possible outcomes. Since there are 52 cards in the deck, there are 52 possible cards to be the 13th card dealt.
Therefore, the probability is calculated as:
Probability = Number of favorable outcomes / Number of possible outcomes
Probability = 4 / 52
Probability = 1 / 13
So, the probability that the 13th card is the first king to be dealt is 1/13.
How long will it take a $800 investment to be worth $900 if it is continuously compounded at 11% per year?
It takes approximately 1.083 years, or around 1 year and 1 month for an $800 investment to grow to $900 when continuously compounded with an 11% yearly interest rate.
Explanation:To calculate how long it will take for an $800 investment to grow to $900 with continuous compounding at an 11% yearly interest rate, we use the formula for continuously compounded interest, P = Pert, where P is the final amount, Pe is the original principal, r is the interest rate, and t is time in years.
Rearranging the equation to find t, we have, t = ln(P/Pe) / r.
Substitute P as $900, Pe as $800, and r as 0.11 into the formula.
Therefore, t = ln(900/800) / 0.11 = approximately 1.083 years or around 1 year and 1 month.
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The number of chicken legs depends on how many chickens are in the coop. This is modeled by y=2x and is an example of _____ variation.
Answer:
y=2x is an example of direct variation.
Step-by-step explanation:
Direct variation is mathematical relationship in between two variables which can be expressed by an equation where one variable is equal to a constant multiplied by the other variable.
In the equation y=2x, x and y are the variables and 2 is a constant.
Here it is expressed as an equation where the variable y is equal to the constant 2 multiplied by the other variable x.
Thus we can say that it is an example of direct variation.
A standard deck of cards contains 52 cards. one card is selected from the deck. ?(a) compute the probability of randomly selecting aa heartheart or spadespade. ?(b) compute the probability of randomly selecting aa heartheart or spadespade or clubclub. ?(c) compute the probability of randomly selecting aa twotwo or diamonddiamond.
The probability of randomly selecting a heart or a spade is 1/2. The probability of randomly selecting a heart or a spade or a club is 3/4. The probability of randomly selecting a two or a diamond is 17/52.
Explanation:(a) There are 13 hearts and 13 spades in a standard deck of cards. The probability of randomly selecting a heart or a spade can be calculated as:
P(Heart or Spade) = P(Heart) + P(Spade) = 13/52 + 13/52 = 26/52 = 1/2
(b) To find the probability of randomly selecting a heart or a spade or a club, we add the probability of selecting a club to the previous result:
P(Heart or Spade or Club) = P(Heart or Spade) + P(Club) = 1/2 + 13/52 = 39/52 = 3/4
(c) There are 4 twos and 13 diamonds in a standard deck of cards. The probability of randomly selecting a two or a diamond can be calculated as:
P(Two or Diamond) = P(Two) + P(Diamond) = 4/52 + 13/52 = 17/52
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Lowell bought 4 1/4 pounds of apples and 3 3/5 pounds of oranges. How many pounds of fruit did lo buy?
Answer:
7.85 pounds (or 7 17/20 pounds)
Step-by-step explanation:
In decimal, the total is ...
4.25 + 3.60 = 7.85 . . . pounds
As mixed numbers, the total is ...
(4 +1/4) + (3 +3/5) = (4 +3) + (1/4 +3/5) = 7 +(5/20 +12/20) = 7 17/20 . . . pounds
Lowell bought 7 17/20 pounds of fruit.
_____
Comment on the math
Any calculator can add these numbers for you. You may have to enter them as (4+1/4) and (3+3/5) if your calculator does not support mixed number entry. Depending on the calculator, you may be able to get it to show you the mixed number result. If you want a mixed number and you get a decimal number, you may have to reduce the fraction yourself:
7 85/100 = 7 17/20 . . . . . . a factor of 5/5 can be removed from the fraction
___
Any two fractions can always be added using the formula ...
a/b + c/d = (ad +bc)/(bd)
Here, that means the sum of 1/4 and 3/5 is ...
1/4 + 3/5 = (1·5 +4·3)/(4·5) = (5+12)/20 = 17/20 . . . . . the result above
In some cases, you may have to reduce the resulting fraction.
Lowell bought [tex]4 \frac{1}{4}[/tex] pounds of fruit.
Explanation:To find the total weight of fruit Lowell bought, we need to add the weights of the apples and oranges together.
Lowell bought [tex]4 \frac{1}{4}[/tex] pounds of apples and [tex]3 \frac{3}{}[/tex] pounds of oranges. To add these fractions, we need to find a common denominator, which in this case is 20.
Converting both fractions to twentieths, we have [tex]\frac{17}{20}[/tex] pounds of apples and [tex]\frac{68}{20}[/tex] pounds of oranges. Adding these together, we get a total of [tex]\frac{85}{20}[/tex] pounds. Simplifying the fraction, we find that Lowell bought [tex]4 \frac{1}{4}[/tex] pounds of fruit.
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Hans's vegetable garden has an area of 400 ft2. It is the shape of a rectangle with a length of 40 ft and a width of 10 ft. He has decided to expand the garden to make room for several new varieties of lettuce. The new garden will be a larger rectangle. He plans on making the new length 3 times the current length and the new width 3 times the current width.
To calculate the new area of Hans's expanded garden, multiply the new length by the new width. The new garden will have an area of 3600 ft^2.
The original garden area is 400 ft2, with a length of 40 ft and a width of 10 ft.
To find the new garden's area after expansion:
Calculate the new length: 40 ft * 3 = 120 ft.Calculate the new width: 10 ft * 3 = 30 ft.Multiply the new length by the new width: 120 ft * 30 ft = 3600 ft2.find the poin on the graph 0f f(x)=1-x^2 that are closest to0(0,0)
How many numbers are between 0 and 1/10
Sonya can walk 6 km in 3 hours. If she has to walk 10 km, how much time will it take her?
Derive the quadratic formula from the standard form (ax2 + bx + c = 0) of a quadratic equation by following the steps below.
a.Divide all terms in the equation by a.
b.Subtract the constant (the term without an x) from both sides.
c.Add a constant (in terms of a and b) that will complete the square.
d.Take the square root of both sides of the equation.
e.Solve for x.
Answer:
See below.
Step-by-step explanation:
We are going to take the quadratic formula ax²+bx+c=0
a.Divide all terms in the equation by a.
[tex]\frac{ax^{2} }{a} +\frac{b}{a}x+\frac{c}{a}=0\\\\x^{2} +\frac{b}{a}x+\frac{c}{a} =0[/tex]
b.Subtract the constant (the term without an x) from both sides.
[tex]x^{2} +\frac{b}{a}x+\frac{c}{a} -\frac{c}{a} =\frac{-c}{a} \\\\x^{2} +\frac{b}{a}x=-\frac{c}{a}[/tex]
c.Add a constant (in terms of a and b) that will complete the square.
[tex]x^{2} +\frac{b}{a}x=-\frac{c}{a}\\\\[tex]x^{2} +\frac{b}{a}x+\frac{b^{2} }{4a^{2} } =-\frac{c}{a} +\frac{b^{2} }{4a^{2}}\\\\(x+\frac{b}{2a}) ^{2} =\frac{-4ac+b^{2} }{4a^{2} }[/tex]
d.Take the square root of both sides of the equation.
[tex]x+\frac{b}{2a} =[tex]\\x+\frac{b}{2a} =\frac{\sqrt{-4ac+b^{2} } }{2a}\\x=\frac{\sqrt{-4ac+b^{2} } }{2a}-\frac{b}{2a} [/tex]
e.Solve for x.
[tex]x=-b+\frac{\sqrt{b^{2}-4ac } }{2a}[/tex]
identity the two rational numbers
Trapezoid ABCD is similar to rectangle EFGH. Side CD is proportional to side ___.
On Monday, a local hamburger shop sold a combined total of 556 hamburgers and cheeseburgers. The number of cheeseburgers sold was three times the number of hamburgers sold. How many hamburgers were sold on Monday?
what is the value of -2 x^2 +4y if x = -2 and y =3
The area of Desert A is nine times the area of Desert B. If the sum of their areas is 2,000,000 square miles, find the area of each desert.
f(x)=2× if x<0
|x| if x>0
To be eligible for a basketball tournament, a basketball team must win at least 60% of its remaining games. If the team has 21 game remaining, how many games must the team win to qualify for the tournament?
The basketball team must win at least 13 out of their 21 remaining games to qualify for the tournament, as they need to win at least 60% of the games.
Explanation:To determine how many games a basketball team must win out of 21 to qualify for a tournament, given they must win at least 60%, we perform the following calculation:
First, multiply the total number of remaining games, which is 21, by 60% (or 0.60) to find the minimum number of games they need to win.Next, solve for the number: 21 games × 0.60 = 12.6.Since a team can't win a fraction of a game, they must win at least the next whole number of games, which is 13.Therefore, the team must win at least 13 games out of the remaining 21 to be eligible for the tournament.
If a cow has a mass of 9×1029×102 kilograms, and a blue whale has a mass of 1.8×1051.8×105 kilograms, which of these statements is true?
A.) The blue whale has about 200200 times more mass.
B.)The blue whale has about 500500 times more mass.
C.)The cow has about 200200 times more mass.
D.)There is no way to compare the masses.
Answer:
The blue whale has about 200 times more mass.
what is 8m-3-7m???????
Answer:
i personally think it's 4
Step-by-step explanation:
even tho this was 4 years ago ehhhhhhh
Allen's monthly take-home pay is $3000, and his monthly rent is $750. If both his monthly take-home pay and his rent increase by $200, what percentage of Allen's take-home pay will be used to pay rent?
Answer:
29.68%
Step-by-step explanation:
Allen's monthly take-home pay is $3000
His monthly rent is $750
Now both his monthly take-home pay and his rent increase by $200
So, After increase Allen's monthly take-home pay =$3000+$200= $3200
After increase His monthly rent = $750+200=$950
Now we are supposed to find what percentage of Allen's take-home pay will be used to pay rent
So, percentage = [tex]\frac{\text{Rent after increase}}{\text {Take- home pay after increase}} \times 100[/tex]
= [tex]\frac{950}{3200}\times 100[/tex]
= [tex]\frac{950}{32}[/tex]
= [tex]29.68\%[/tex]
Hence 29.68% of Allen's take-home pay will be used to pay rent.