A football is kicked off the ground. After traveling a horizontal distance of 24m, it just passes over a tree that is 3m tall before hitting the ground 28m from where it was kicked. a) Consider the ground to be the x-axis and assume the vertex lies on the y-axis. Determine the equation of the quadratic equation that models the path of the ball. b)Determine the maximum height of the ball. c)Suppose the football is kicked from a starting point at the origin. Develop a new equation of the quadratic function of the quadratic function that represents the football.
Answers
Answers:
a) [tex]y = -\frac{1}{32} x^2 + \frac{49}{8}[/tex]
b) [tex] \frac{49}{8} \text{ meters} [/tex]
c) [tex]y = -\frac{1}{32} (x-14)^2 + \frac{49}{8}[/tex]
Explanations:
a) Note that the path where the ball travels when kicked off the ground is the parabola.
Let (h,k) be the vertex of the parabola. Since the vertex is in y-axis, the x-coordinate of the vertex is 0. So, h = 0.
The equation of the parabola with vertex (h, k) is given by
[tex]y = a(x - h)^2 + k[/tex] (1) Since h = 0, we can replace this to equation (1) so that
[tex]y = a(x - h)^2 + k\\ y = a(x - 0)^2 + k\\ y = ax^2 + k[/tex] (2)
To solve for a and k, we need to get the coordinates of the points of parabola. In the problem, it is possible to determine the coordinates of the points where the ball is kicked off from the ground and where the ball ended hitting the ground and the point where the ball hits the tree which are points of the parabola.
Since the ground is in x-axis and the ball is kicked off from the ground and ended hitting the ground, the parabola intersects the x-axis at the points where the ball is kicked off from the ground and where the ball ended hitting the ground.
Let
A = point where the ball is kicked off from the ground
B = point where the ball ended hitting the ground
M be the midpoint of A and B.
Note that the x-coordinate of the vertex of the parabola is the same as the x-coordinate of M. Since the x-coordinate of the vertex of the parabola is 0, the x-coordinate of M is 0. Moreover, since the ground is in x-axis, A and B are in the x-axis and so M is in x-axis. So, the y-coordinate of M is 0 because the y-coordinate of any points of x-axis is 0. Thus, the coordinates of M are (0,0).
As mentioned earlier, the y-coordinate of any points in x-axis is 0. Since A and B both lie in x-axis, their y-coordinate is 0. Now, we let j be the x-coordinate of A and k be the x-coordinate of B so that (j, 0) are the coordinates of A and (k, 0) are the coordinates of B.
Since the ball traveled with a horizontal distance of 28 meters when it is kicked off until it hit the ground, the horizontal distance from A to B is 28. Since the coordinates of A and B are (j, 0) and (k, 0) respectively, using the distance formula the distance from A and B is given by
[tex]d = \sqrt{(j-k)^2 + (0 - 0)^2} \\ 28 = \sqrt{(j-k)^2} \\ (j-k)^2 = 28^2 \\ \boxed{j - k = \pm 28}[/tex]
Since A is the point where the ball is kicked off and B is the point where the ball hits the ground, A is in the left of B. Since A and B are both in x-axis and j and k are the x-coordinates of A and B respectively, j < k. Thus, j - k is negative and so
j - k = -28 (3)
Because M is the midpoint of A and B, the x-coordinate of M is [tex] \frac{j+k}{2}[/tex]. Since the x-coordinate of M is 0,
[tex]\frac{j+k}{2} = 0[/tex] (4)
Multiplying both sides of equation (4) by 2, we have
j + k = 0 (5)
Now, we add equations (3) and (5) so that
2j = -28
j = -14
Substituting the value of j in (5), we have
j + k = 0
-14 + k = 0
k = 14
Hence the coordinates of A are (-14,0) and the coordinates of B are (14,0). Since point A is in parabola, we can substitute its coordinates of A in equation (2) with x = -14 and y = 0 so that
[tex]y = ax^2 + k \\ 0 = a(-14)^2 + k \\ 0 = 196a + k \\ \boxed{196a + k =0} [/tex] (6)
Note that, if we substitute the coordinate of B, we also arrive at equation (6), which is useless.
So, we need to determine the coordinates of another point of parabola, which is the point where the ball hits the tree. We call this point C. Since the tree is 3 meters high and the ground is in x-axis, the y-coordinate of point C is 3. Let c be the x-coordinate of C. Since the ball hits the tree after travelling a horizontal distance of 24 meters, the horizontal distance from A to C is 24. Moreover, the horizontal distance is the difference of x-coordinates of two points. Since C is the right of A and the x-coordinate of A is -14
c - (-14) = 24
c + 14 = 24
c = 10
Hence, the coordinates of C are (10, 3). Substituting this to equation (2), we have
[tex]y = ax^2 + k \\ 3 = a(10)^2 + k \\ 3 = 100a + k \\ \boxed{100a + k =3} [/tex] (7)
Subtracting equation (6) to (7), we have
[tex](196a + k)-(100a + k) = 0-3 \\ 96a = -3 \\ \boxed{a = -\frac{1}{32} }[/tex]
Substituting the value of a in equation (6) we have
[tex]196a + k = 0 \\ 196(-\frac{1}{32}) + k = 0 \\ \\ -\frac{49}{8} + k =0 \\ \\ \boxed{k = \frac{49}{8} } [/tex]
Using equation (2) and the values of a and k, the equation of the parabola is given by
[tex]\boxed{y = -\frac{1}{32} x^2 + \frac{49}{8} }[/tex]
Related Questions
WILL GIVE BRAINLIEST TO PERSON WHO ANSWERS CORRECTLY!!!!!!!!!
Tony is baking cakes and loaves of bread for a family reunion. His cake recipe uses 5 cups of flour for each cake, and his bread recipe uses 6 cups of flour for each loaf. He has 90 cups of flour on hand.
Let c represent the number of cakes and b represent the number of loaves of bread that Tony bakes.
Create an inequality that symbolizes this situation, and then answer the question below.
Can Tony bake 8 cakes and 9 loaves of bread without buying more flour?
Answers
5*6=30*3=90 YES
is 4 yards bigger than 13 feet
Answers
4 yards x 3 feet = 12 feet
12 feet is less than 13 feet
So False, 4 yards is less than 13 feet
hope this helps
Here is your answer:
The proper answer to this question is "no".
1 yard=3 feet
4*3=12
12<13
Your answer is no.
If you need anymore help feel free to ask me!
Hope this helps!
Describe two different sequences of transformations that will map ABCD to EFGH.
Answers
Translation before rotation:
.. translate down 5 units
.. rotate counterclockwise 90° about the origin
Translation after rotation:
.. rotate counterclockwise 90° about the origin
.. translate right 5 units
This can also be accomplished by a double reflection.
.. reflect across the line y=x
.. reflect across the line x=2.5
Or,
.. reflect across the line y=2.5
.. reflect across the line y=x
Answer:
The orientation of the figure has been changed by 90°, but the points are not all rotated around the same center. Translation is involved.
Translation before rotation:
.. translate down 5 units
.. rotate counterclockwise 90° about the origin
Translation after rotation:
.. rotate counterclockwise 90° about the origin
.. translate right 5 units
This can also be accomplished by a double reflection.
.. reflect across the line y=x
.. reflect across the line x=2.5
Or,
.. reflect across the line y=2.5
.. reflect across the line y=x
Step-by-step explanation:
please mark this answer as the brainlest
Describe the relationship that must exist between a, b, and c in the equation ax^2+bx+c=0 in order for the equation to have exactly one solution
Answers
The [tex]b^2-4ac[/tex] bit is called the discriminant and it tells you what kind of solutions occur,
-if [tex]b^2-4ac \ \textgreater \ 0[/tex] then the equation has two different solutions
-if [tex]b^2-4ac=0[/tex] then the equation has exactly one solution
-if [tex]b^2-4ac\ \textless \ 0[/tex] then the equation has no (real) solutions.
So you are interested in exactly one solution which happens if [tex]b^2-4ac=0[/tex] or equivalently [tex]b^2=4ac[/tex].
A graduated cylinder is filled with 36π of liquid. The liquid is poured into a different cylinder that has a radius of 3 cm. What will the height of the liquid be in the new cylinder?
Answers
Answer:
The height of the liquid in the new cylinder = 4 cm
Step-by-step explanation:
∵ The volume of the liquid = 36π cm³
∵ It poured into a cylinder that has radius 3 cm
∵ The volume of any cylinder = πr²h
∴ The volume of the liquid in the cylinder = π(3)²h = 9πh
- Where h is the height of the liquid inside the cylinder
∴ 9πh = 36π ⇒ divide both sides by π
∴ 9h = 36 ⇒ h = 36/9
∴ h = 4 cm
∴ The height of the liquid in the new cylinder = 4 cm
The height of the liquid in the new cylinder will be 4 cm. The term height is frequently used to describe someone's or something's height.
What is the height?Height is defined as the vertical distance between an object's top and bottom. It is measured in centimeters, millimeters, meters, and other units.
From the given conditions;
The volume of the liquid is, [tex]\rm V_L=36 \pi \ cm^3[/tex]
The radius of the cylinder is,[tex]\rm r=3 cm[/tex]
The volume of the cylinder is found as;
[tex]\rm V= \pi r^2 h \\\\ \rm V= \pi (3)^2 h \\\\ \rm V=9\pi h[/tex]
The height of the liquid is in the new cylinder is found as;
The volume of cylinder = Volume of liquid
[tex]\rm 9 \pi h= 36 \pi \\\\ \rm h=\frac{36}{9} \\\\ h= 4 \ cm[/tex]
Hence the height of the liquid in the new cylinder will be 4 cm.
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The equation of a circle whose center is at (4, 0) and radius is length 2√(3) is (x - 4)² + y² = 2√3 (x - 4)² + y² = 12 (x + 4)² + y² = 12
Answers
(x-4)²+(y-0)²=(2√3)²
(x-4)²+y²=12
Answer:
Equation of the circle is
[tex](x-4)^2+y^2=12[/tex]
Step-by-step explanation:
We have been given that
center of the circle = (4,0)
Radius of the circle = [tex]2\sqrt3[/tex]
The standard form of a circle is given by
[tex](x-h)^2+(y-k)^2=r^2[/tex]
Here, (h,k) is the center and r is the radius. Thus, we have
h = 4, k = 0, r = 2√(3)
Substituting these values in the above equation, we get
[tex](x-4)^2+(y-0)^2=(2\sqrt3)^2[/tex]
Simplifying, we get
[tex](x-4)^2+y^2=12[/tex]
Therefore, equation of the circle is
[tex](x-4)^2+y^2=12[/tex]
(I NEED THIS ANSWERED IMMEDIATELY. IT'S WORTH 15 POINTS!!!!!
Write a ratio the situation in three ways comparing the first quantity to the second quantity. A zoo has 14 monkeys and 18 chimpanzees.
Answers
A zoo has 14 monkeys and 18 chimpanzees.
To write the ratio of numbers x and y we use three ways given below
1. x to y
2. x : y
3. [tex] \frac{x}{y} [/tex]
Now write the ratio in three ways comparing 14 monkeys and 18 chimpanzees.
Let x be the number of monkeys
y be the number of chimpanzees
So ratio becomes
14 to 18
14 : 18
[tex] \frac{14}{18} [/tex]
Randall is going to the grocery store. It is 3 3/5 miles from his house. He walks 2 2/5 miles before it starts to rain. How far will he have to walk in the rain?
Answers
which answer is a reasonable estimate to the following problem? 94.689
Answers
Devin borrowed $1,058 at 13 percent for nine months.
What will he pay in interest?
How much is his total payment?
What will be his monthly payment?
Answers
Answer:
A. [tex]I\approx \$103.16[/tex]
B. [tex]\text{Total payment}=\$1161.16[/tex]
C. [tex]\text{The monthly payment}\approx \$129.02[/tex]
Step-by-step explanation:
We have been given that Devin borrowed $1,058 at 13 percent for nine months.
A. To find the amount of interest paid by Devin we will use formula: [tex]I=Prt[/tex], where,
[tex]I=\text{Amount of interest}[/tex],
[tex]P=\text{Principal amount}[/tex],
[tex]r=\text{Interest rate in decimal form}[/tex],
[tex]t=\text{Time in years}[/tex].
Let us convert our given interest rate in decimal form and time in years.
[tex]13\%=\frac{13}{100}=0.13\\\text{9 months}=\frac{9}{12}\text{ year}=\frac{3}{4}=0.75\text{ year}[/tex]
Upon substituting our given values in interest formula we will get,
[tex]I=\$1058*0.13*0.75[/tex]
[tex]I=\$103.155\approx \$103.16[/tex]
Therefore, the amount of interest paid by $103.16.
B. Since total payment will be equal to interest rate plus amount of interest.
[tex]\text{Total payment}=\$1058+\$103.16[/tex]
[tex]\text{Total payment}=\$1161.16[/tex]
Therefore, Devin's total payment will be $1161.16.
C. To find the monthly payment we will divide total amount by 9.
[tex]\text{The monthly payment}=\frac{\$1161.16}{9}[/tex]
[tex]\text{The monthly payment}=\$129.01722\approx \$129.02[/tex]
Therefore, the amount of Devin's monthly payment is $129.02.
A hacker is trying to guess someone's password. the hacker knows (somehow) that the password is 6 digits long, and that each digit cound be a number between 0 and 8. assume that the hacker makes random guesses. what is the probability that the hacker guesses the password on his first try? round to four decimal places.
Answers
The probability that the hacker guesses the password on his first try is 0
How to determine the probability?The password is given as:
Password = 6 digit long
The usable numbers are:
Digits = 0 to 8 i.e. 9 digits
If the digits can be repeated, then the number of passwords is:
Password = 9^6
Evaluate the exponent
Password = 531441
The probability that the hacker guesses the password on his first try is
P = 1/531441
Evaluate
P = 0.00000188167
Approximate
P = 0.0000
Hence, the probability that the hacker guesses the password on his first try is 0
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The probability of guessing the password on the first try is 0.000001881.
Explanation:To calculate the probability of guessing the password on the first try, we need to determine the number of possible passwords and divide it by the total number of possible guesses. In this case, the password is 6 digits long and each digit can be a number between 0 and 8.
So, there are 9 possible digits for each of the 6 positions, giving us a total of 9^6 = 531441 possible passwords.
The hacker makes random guesses, so there are 9^6 = 531441 possible guesses they can make.
The probability of guessing the password on the first try is therefore 1/531441 = 0.000001881, rounded to four decimal places.
Peyton has 2 pizzas. Each pizza is cut into 10 equal slices. She and her friends eat 14 slices. What part of the pizzas did they eat?
Answers
70% part of the pizzas they ate.
What is percentage?A value or ratio that may be stated as a fraction of 100 is referred to as a percentage in mathematics. If we need to compute a percentage of a number, we should divide it by its whole and then multiply it by 100. The proportion therefore refers to a component per hundred. Per 100 is what the term percent signifies. The letter "%" stands for it.
Given, Peyton has 2 pizzas. Each pizza is cut into 10 equal slices.
Thus, total slices of pizza = 10 *2
total slices of pizza = 20
Since, She and her friends eat 14 slices.
Thus,
Percentage of pizza they eat = 14/20*100
Percentage of pizza they eat = 14*5
Percentage of pizza they eat = 70%
Therefore, Alex and her friends Eat 70% of the pizza.
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evaluate r s + 14s when r= 6 and s= 1/4
Answers
(6)(1/4) + 14(1/4)
1.5 + 14(1/4)
1.5 + 3.5
5
The common denominator of 3/4 and 9/10
Answers
So 3/4 would be 15/20, and 9/10 would become 18/20.
What is the ninth term in the binomial expansion of (x – 2y)13? 329,472x5y8 –329,472x5y8 –41,184x8y5 41,184x8y5
Answers
Please, see the detailed solution in the attached file.
Thanks.
The necklace charm shown has two parts, each shaped like a trapezoid with identical dimensions.
What is the total area, in square millimeters, of the charm?
Answers
Answer:
855mm
Step-by-step explanation:
The area of the trapezoid can be calculated through the equation,
A = (b₁ + b₂)h / 2
where b₁ and b₂ are the bases and h is the height. Substituting the known values from the given,
A = (25mm + 32mm)(15 mm) / 2
A = 427.5 mm²
Since there are two trapezoids in the necklace, the area calculated is to be multiplied by two to get the total area.
total area = (427.5 mm²)(2)
total area = 855 mm²
A bag contains 15 red candies, 20 blue candies, and 10yellow candies. What is the probability of drawing a red candy? A blue candy? Something besides a yellow candy?
Answers
The probability of drawing a red, blue, and a candy that is not yellow from the bag is 1/3, 4/9, and 7/9 respectively.
Explanation:The question involves the concept of probability in mathematics, with a scenario of drawing candies from a bag. The total number of candies in the bag is 45 (15 red + 20 blue + 10 yellow).
The probability of drawing a red candy is the number of red candies divided by the total number of candies, which is 15/45 or 1/3.
The probability of drawing a blue candy is the number of blue candies divided by the total number of candies, which is 20/45 or 4/9.
If we want to find the probability of drawing something other than a yellow candy, this is essentially the sum of the probabilities of drawing a red or blue candy. This is 35 out of 45 (15 red candies + 20 blue candies), which reduces to 7/9.
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Solve for W!
W - 2.76 = 6.7
Answers
isolate the W by adding 2.76 to both sides of the equal sign
W - 2.76 (+2.76) = 6.7 (+2.76)
W = 6.7 + 2.76
W = 9.46
9.46 is your answer
hope this helps
W= 9.46
Kevin rented a truck for obe day. There was a base fee of $15.99, and there was an additional charge of 83 cents for each mile driven. Kevin had to pay $177.01 when he returned the trunk. For how many miles did ge drive the trunk?
Answers
This should give you 161.02
Then divide 161.02 by .83
He drove 194 miles :)
Please help me with questions 14 and 15
Answers
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Problem 15: The answer is choice A
cos(angle) = adjacent/hypotenuse
cos(B) = BC/AB
cos(B) = 12/13
what is the explicit rule for the following sequence -10 - 12 -14 -16
Answers
The sum of two numbers is 46 and the difference is 14. What are the numbers?
Answers
.. x -(46 -x) = 14 . .the difference is 14
.. 2x = 60 . . . . . . simplify
.. x = 30 . . . . . . . divide by 2
The larger number is 30; the smaller is 16.
x + y = 46
x - y = 14 Add these two equations.
2x = 60 Divide by 2.
x = 60/2
x = 30
x + y = 46 Put 30 in for x
30 + y = 46 Subtract 30
y = 36 - 30
y = 16
the larger number is 30. The smaller is 16
Factor
12x^5+6x^2-18
A) 6x^2(2x^3+x−3)
B) 12(x^5+6x^2−18)
C) 6(2x^5+x^2−3)
D) 6(2x^5+6x^2−18)
Answers
6(x-1)(2x^4+2x^3+2x^2+3x+3)
I hope this helped but I couldn't see this as one of the multiple choice answers but I worked it out and this I what I got
Graph the following inequality. Then click until the correct graph appears.
x - y < 2
Answers
the correct graph is the dotted diagonal one ( 3rd picture)
Solve for y
y >x-2
Answer:
Third picture
Step-by-step explanation:
If you simplify the inequality, it is y > x - 2. Because of this, we can graph it as such.
Assume Triangle EFG equals triangle JKL
Answers
Since ΔEFG is congruent to ΔJKL, their corresponding angles are congruent (CPCTC). The congruence statement tells you which angles of the triangle correspond; ∠E corresponds to ∠J, ∠F corresponds to ∠K, and ∠G corresponds to ∠L. The measures of all the angles of ΔJKL are given, so you just have to name the angle of ΔJKL that corresponds to ∠E of ΔEFG.
Looking at the congruence statement, ∠J corresponds to ∠E, and the m∠J is 90°. Thus the answer is 90°.
Answer:
m∠E = 90°
What is the median of this set of data? 1, 2, 5, 6, 9 1 5 8 9
Answers
A used bookstore will trade 2 of its books foe of yours, If Val brings in 18 books to trade,how many books can she get from the store
Answers
Hope this helps!
♤MidnightQueen
A box of 24 chocolate truffles cost $28.50 and individual truffle cost $1.50 how much do you save by purchasing a box of 24
Answers
[tex]1.50 \times 24 = 36 \\ 36 - 28.50 = 7.5[/tex]
you save $7.50
A smaller number is 3 less than half a larger number. The larger number is 10 times 1 less than the smaller number. Let x represent the smaller number, and let y represent the larger number. Which equations can be used to model the situation? Check all that apply.
Answers
Answer:
A,B,E
Step-by-step explanation:
HAVE A GOOD DAY
The required expressions that represent the given conditions are x = 1/2y - 3 or 2x - y = -6 and y = 10[x - 1]. Options A, B, and E are correct.
Given that,
A smaller number is 3 less than half a larger number. The larger number is 10 times 1 less than the smaller number. Let x represent the smaller number, and let y represent the larger number.
The equation is the relationship between variables and represented as y = ax + b is an example of a polynomial equation.
Here,
According to the question,
A smaller number is 3 less than half a larger number.
x = 1/2y - 3
or
2x -y = -6
The larger number is 10 times 1 less than the smaller number.
y = 10[x - 1]
Thus, the required expressions that represent the given conditions are x = 1/2y - 3 or 2x - y = -6 and y = 10[x - 1]. Options A, B, and E are correct.
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Solve and graph question 5 help with this problem
Answers
6 - 2x > 2x + 26
Let's start by subtracting 2x from both sides
6 - 2x - 2x > 2x - 2x + 26
-4x + 6 > 26
Then add -6 to both sides
-4x + 6 - 6 > 26 - 6
-4x > 20
Divide both sides by -4
-4x/-4 > 20/-4
x < -5
As you can see, I change the sign. The reason is because whenever you divide both sides by a negative sign, you MUST change the sign.
The answer is -5
See picture bellow for the graph!
I hope this answer helps, and please let me know if you have questions about the answer.
As always, I am glad to be your helper to night!