Allie is playing basketball. She takes a shot 24 ft away. when the ball is 4 feet away from her, it is at a height of 10 feet above the floor. the ball reaches its greatest height of 18 ft above the floor when it is 12 feet away from her. a. find the value of a. b. if the hoop is 10 ft high, how close does Allie have to be in order to make the basket?
Answers
Part (a)
Answer: a = -1/8In decimal form, this is a = -0.125
-------------------
Explanation:
When it says "she takes the shot 24 feet away", that means that the horizontal distance from the basket to Allie is 24 feet. It's not measuring the slanted or diagonal distance directly from Allie's hand to the net itself. We can think of it as the distance from her feet to the base of the hoop.
When measuring the distance from Allie to the ball, we use the same idea. We simplify things using horizontal distance. Things get way too complicated if we measure the direct distance from the ball (wherever it may be in the air) to the net, or from the ball to Allie's hand.
x = horizontal distance the ball is away from Alliey = height of the ballx and y have units in feet.
The info "when the ball is 4 feet away from her, it is at a height of 10 feet above the floor" means we know the point (x,y) = (4,10) is on the parabola.
The other piece of info that "the ball reaches its greatest height of 18 ft above the floor when it is 12 feet away from her". This tells us the vertex is (h,k) = (12,18)
We have these four pieces of info
x = 4y = 10h = 12k = 18Plug those values into the equation below to isolate 'a'
y = a(x-h)^2 + k .... vertex form
10 = a(4-12)^2 + 18
10 = a(-8)^2 + 18
10 = 64a + 18
64a + 18 = 10
64a = 10-18
64a = -8
a = -8/64
a = -1/8
This converts to the decimal -1/8 = -0.125
============================================================
Part (b)
Answer: 20 feet-------------------
Explanation:
From part (a), we found a = -1/8 = -0.125
This means that the vertex form equation y = a(x-12)^2 + 18 updates to y = -0.125(x-12)^2 + 18
This parabola represents the path the ball takes. Plug in y = 10 and solve for x. We do this because the hoop is 10 ft high, and we want to know what x values will make the equation true when y = 10. In other words, we want to know what two locations on the parabola in the form (x,10).
So,
y = -0.125(x-12)^2 + 18
10 = -0.125(x-12)^2 + 18
10-18 = -0.125(x-12)^2
-8 = -0.125(x-12)^2
-0.125(x-12)^2 = -8
(x-12)^2 = -8/(-0.125)
(x-12)^2 = 64
x-12 = 8 or x-12 = -8
x = 8+12 or x = -8+12
x = 20 or x = 4
The two solutions are x = 4 and x = 20.
The two points we focus on are (4,10) and (20,10). This is when the ball is 10 ft off the ground. Earlier we were told that when the ball is 4 ft away from her, the ball is 10 ft off the ground. So that helps confirm why x = 4 works as a solution. We'll ignore this value though because we want the ball to come down through the hoop, not go up through the hoop. See the diagram below.
Therefore, she needs to be 20 ft away from the hoop to make the shot. Her being 24 ft away means that she's 24-20 = 4 ft too far.
Check out the graph below. We have the following points
A = location when the ball is 4 ft away from her, and 10 ft off the groundB = highest point, aka vertexC = second point when the ball is 10 ft off the ground, now the ball is heading downwardD = basket's location, it's x = 24 ft away from Allie, and y = 10 ft off the groundAs you can see in the diagram below, the green arc represents her shot if she's 24 ft away. She misses point D (the hoop). To make the hoop, she needs to move 4 ft closer to the basket. The red arc is the path the ball takes that will land in the hoop. Every point on the green arc has been shifted over 4 units to the right to arrive at a corresponding point on the red arc.
The vertex form of the parabola can be given as,
\(y=a(x-h)^2+k\)
a)The value of \(a\) is -1/8.b)If the hoop is 10 ft high, the distance close to Allie have to be in order to make the basket should be 20 feet.What is the vertex form of the parabola?The vertex form of the parabola can be given as,
\(y=a(x-h)^2+k\)
Given information-
Allie takes a shot of 24 ft away.
At the distance of 4 feet the ball achieve the height of 10 feet from the ground.
The greatest height achieved by the ball is 18 ft above the floor.
a) The value of a.The vertex form of the parabola can be given as,
\(y=a(x-h)^2+k\)
Here, \((h,k)\) are the vertex of the parabola.
As the ball was 4 feet away from Allie, it is at a height of 10 feet above the floor. Thus the distance on the coordinate on x axis should be 4 feet and The distance on the y axis should be 10 feet.
Now when the ball is 12 feet away the height is 18 ft. Thus the vertex of the parabola is (12,18).
Put the values in the above equation as,
\(10=a(4-12)^2+18\\10=a(-8)^2+18\\10-18=64a\\a=-\dfrac{2}{64} \\a=-\dfrac{1}{8}\)
Thus the value of \(a\) is -1/8.
b) If the hoop is 10 ft high, The distance close to Allie have to be in order to make the basket-Now the value of y is 10 and x is not known.
All the other values are same. Put these values in the vertex form, we get,
\(10=-\dfrac{1}{8} (x-12)^2+18\\\\10-18=-\dfrac{1}{8}(x-12)^2\\(x-12)^2=8\times 8 \\(x-12)^2=64\\(x-12)=\pm8\)
By taking positive sign and negative sign one by one the values of x occurs 20 and 4.
Thus the two points we get are (4,10) and (20,10).
Thus If the hoop is 10 ft high, the distance close to Allie have to be in order to make the basket should be 20 feet.
Hence,
a)The value of \(a\) is -1/8.b)If the hoop is 10 ft high, the distance close to Allie have to be in order to make the basket should be 20 feet.Learn more about the vertex form of the parabola here;
https://brainly.com/question/17987697
Related Questions
what is the rule dividing integers with different signs
Answers
Which table does not represent a proportional relationship?
Answers
Answer:
Table 3
Step-by-step explanation:
The ratio of x and y is not constant.
What are all the whole numbers that make 8
- ? > 3 true
0, 1, 2, 3, 4,5
0, 1, 2, 3, 4
0, 1, 2
5
Answers
Answer:0,1,2,3,4
Step-by-step explanation:
If you solve it by doing 8-x>3, then you get x<5.
That means all numbers that are less than 5, and if we try that with the numbers given, it’s true. So therefore it’s 0,1,2,3,4
Hope this helps!
Which of the following statements justifies why the triangle shown below is not a right triangle?
Z
O A. 42 +122132
OB. YZ + XZ > XY
OC. XZ
O D. YZ
SUBMIT
please help
Answers
None of the options are correct. And the given triangle is a right-angle triangle.
Given that,
Which of the following statements justifies why the triangle shown below is not a right triangle is to be determined.
The triangle is a geometric shape that includes 3 sides and the sum of the interior angle should not be greater than 180°
Here,
A triangle that follows the Pythagoras theorem, are right an angles triangle,
For the given triangle,?
13² = 4² + 12²
169 = 25 + 144
169 = 169
So, the given triangle holds the property of Pythagoras' theorem,
Thus, None of the options are correct. And the given triangle is a right-angle triangle.
Learn more about triangles here:
https://brainly.com/question/2773823
#SPJ2
R(9, 7) and S(3,9) are the endpoints of a line segment. What is the midpoint M of that line
segment?
Write the coordinates as decimals or integers.
M=
Answers
Answer:
MP=(6,8)
Step-by-step explanation:
\(MP= (\frac{x_1+x_2}{2} )(\frac{y_1+y_2}{2} )\\MP= (\frac{9+3}{2} )(\frac{7+9}{2})\\MP= (\frac{12}{2} )(\frac{16}{2} )\\MP= (6,8)\)
A quadratic equation with opposite solutions can be found by multiplying
left parenthesis x minus r right parenthesis and left parenthesis parenthesis____
The equation will have ____
Answers
A quadratic equation with opposite solutions can be found when D > 0.
In a quadratic equation, the discriminant D describes the nature of the roots which is also called the solutions of the equation.
For a quadratic equation ax²+bx+c = 0, the discriminant D is given by =
D = [-b ± √(b² – 4ac)]/2a
Nature of roots:
D > 0, roots are real and distinct.
D = 0, roots are real and equal.
D < 0, roots are imaginary and unequal.
Hence, the quadratic equation with opposite solutions can be found when D > 0.
Learn more about quadratic equation click;
https://brainly.com/question/30098550
#SPJ4
question 10 a data analyst creates two different predictive models for the same dataset. they use the bias() function on both models. the first model has a bias of -40. the second model has a bias of 1. which model is less biased?
Answers
The model with a bias of 1 is less biased compared to the model with a bias of -40.
Bias in the context of predictive models refers to the tendency of a model to consistently underpredict or overpredict the target variable. A negative bias indicates that the model consistently underpredicts the target, while a positive bias suggests that the model consistently overpredicts the target.
In this case, the first model with a bias of -40 indicates that, on average, its predictions are 40 units lower than the actual values of the target variable. This suggests a systematic underprediction by the model.
On the other hand, the second model with a bias of 1 implies that, on average, its predictions are only 1 unit higher than the actual values of the target variable. This indicates a much smaller deviation from the actual values and suggests a lower level of systematic error compared to the first model.
Therefore, based on the given information, the model with a bias of 1 is less biased compared to the model with a bias of -40. However, it's important to note that bias alone is not the only factor to consider when evaluating the performance of predictive models.
Other metrics, such as accuracy, precision, and recall, should also be taken into account to have a comprehensive understanding of the models' performance and effectiveness.
For more question on bias visit:
https://brainly.com/question/30461915
#SPJ8
2a2 + 3b2 + a2 + b2
Answers
Answer:
Solution given:
2a2 + 3b2 + a2 + b2
adding like terms
2a²+a²+3b²+b²
3a²+4b² is a required answer.
Hailey is putting up a fence in the shape of an isosceles triangle in her
backyard. The fence has side lengths as shown, where x represents the
number of feet in each fence section. The perimeter of the fence can be
covered using 8 total fence sections represented by the expression 8x. If
fencing costs $6.50 a foot, what would be the total cost of the fence?
(2x + 4) ft
(3x + 7) ft
.
(2x + 4) ft
Answers
Answer:
the total cost would be $ 780.00
Step-by-step explanation:
step by step explanation
Suppose you flip a coin five times. What is the probability of obtaining five tails in a row assuming the coin is fair?.
Answers
Answer: 50%
Step-by-step explanation:
The answer is 50%.
determine an equation of a line in standard form that is perpendicular lo the line 2y = -x + 3 and passes through the point (4, 5).
Answers
Mr. Russo had 6.25 cubic feet of potting soil in his raised garden bed. Then he took a bag of soil and poured in enough to completely fill the 9-cubic-foot bed. Let x represent how much soil was in the bag. Which inequality describes the problem?
Answers
Answer:
b
Step-by-step explanation:
A house on the market was valued at $478,000. After several years, the value increased by 8%. By how much did the house's value increase in dollars? What is
the current value of the house?
Answers
Answer:
$38240 increase$516240 current valueStep-by-step explanation:
Initial value of house is $478000
Change in value is 8% greater:
478000 + 8% = 478000 + 8*478000/100 = 478000 + 38240 =516240We need 8% of Value
\(\\ \sf\longmapsto 8\%\:of\;478000\)
\(\\ \sf\longmapsto 0.08(478000)\)
\(\\ \sf\longmapsto 38240\$\)
Current value:-
\(\\ \sf\longmapsto 478000+38240=516240\)
A rice cooker costs $3m. A kettle is $35 cheaper. Find the cost of the kettle.
(Give your answer in terms of m.)
Pls answer
Answers
Answer:
2,999,875
Step-by-step explanation:
3,000,000m -35=2,999,875
=$2,999,965
=$2.999965m
PLEASE SHOW WORK
HELP
Answers
Answer:
82 + 0.6h = 85 and h = 5
Step-by-step explanation:
hours = h
82 + 0.6h = 85
85 - 82 = 0.6h
3 = 0.6h
3/0.6 = h
h = 5
What is this expression in simplified form? 5v2 - 9v6
Answers
A n s w e r:
v^2(5-9v^4)
Someone please help me i will mark brainiest !!!!!
What do you know to be true about the values of a and b ?
A. a = b
B. a < b
C. a > b
D. Cant be determined
Answers
The answer to this question is B
the measure of two vertical angles are 8x-18 and 6x+14 find x
A:16
B:110
C:32
D:70
18+14=32
32 divided by 2 = 16
is this right??
Answers
When the measure of two vertical angles are 8x-18 and 6x+14, x is A. 16.
How to illustrate the information?It should be noted that vertically opposite angles are equal. In this case, we have to equate both angles given. This will be illustrated as:
Angle 1 = Angle 2
8x - 18 = 6x + 14
Collect like terms
8x - 6x = 14 + 18
2x = 32
Divide
x = 32/2
x = 16
The value of x is 16.
Learn more about angles on:
brainly.com/question/25716982
#SPJ1
in your own words explain angle addition postulate
Answers
This postulate tells us that when we put two angles side by side with a ray touching the ray of the other angle, we are creating a new angle which is the addition of the two, and whose measure is the addition of the measures of the two angles we are contacting via a common ray.
What value of x makes the equation true?
Answers
Answer:
B.2.5
Step-by-step explanation:
The equation would be
\(-4x +8 = 2x -7\)
I believe that you would know how to solve this.
You could also sub in 2.5 for the x to check.
Good Luck!!!!!
will give 15 points if the question is answered
Answers
The surface area of a cylinder is equal to 208π square units.How to calculate the surface area of right cylinder?The surface area is the sum of the areas of all faces of the solid. In the case of a right cylinder, the surface area is the sum of the two areas of the circular base and the lateral area (A), in square units, whose formula is described below:A = 2π · r² + 2π · r · h (1)Where:r - Base radiush - HeightIf we know that r = 8 and h = 5, then the surface area of the right cylinder is:A = 2π · 8² + 2π · 8 · 5A = 208πThe surface area of a cylinder is equal to 208π square units.
Type the correct answer in each box. Use numerals instead of words. If necessary, use / for the fraction bar(s).
4(px+1)=64
what is the value of x in terms of p
what is the value of x when p is -5
Answers
Answer:
x = 15/px = -3Step-by-step explanation:
Given expression
4(px+1)=64Solving for x
4px + 4 = 644px = 60px = 15x = 15/pValue of x when p= - 5
x = 15/(-5) = -3Answer:hdhdh
Step-by-step explanation:
algorithm depth-first search : find the order of visit: 2. algorithm breadth first search : find the order of visit
Answers
The order of visit for DFS and BFS will differ based on the structure of the graph or tree and the starting node. DFS will typically visit nodes in a deeper order, while BFS will visit nodes in a breadth-first order.
Both algorithm depth-first search and algorithm breadth-first search are used to traverse or search a graph or tree data structure. The difference between them lies in their traversal strategy.
Depth-first search (DFS) starts at a root node and explores as far as possible along each branch before backtracking. It follows a content-loaded algorithm, meaning that it prioritizes exploring deeper into the structure before considering other branches. In terms of visit order, DFS starts at the root node, then moves to its first child node, and continues to explore its descendants until it reaches a leaf node. Once it has visited all descendants of the current node, it backtracks to the previous node and repeats the process until it has visited all nodes.
On the other hand, breadth-first search (BFS) starts at the root node and explores all the neighboring nodes at the current depth level before moving on to the next depth level. It follows a breadth-first algorithm, meaning that it prioritizes exploring all nodes at a given depth level before moving on to deeper levels. In terms of visit order, BFS visits all nodes at a given depth level before moving on to the next level.
learn more algorithm about here:
https://brainly.com/question/28724722
#SPJ11
What is the Hardy-Weinberg equation and what does each part mean?
Answers
The Hardy-Weinberg equation is a mathematical formula used to calculate the frequency of alleles (alternative forms of genes) in a population.
It states that the frequency of alleles and genotypes in a population will remain constant from generation to generation in the absence of other evolutionary influences.
The equation is expressed as: p² + 2pq + q² = 1
In this equation, p represents the frequency of the dominant allele in the population, and q represents the frequency of the recessive allele.
The superscripts 2 represent the proportion of individuals in the population that are homozygous for that allele (meaning they have two copies of the same allele), while the 2pq term represents the proportion of individuals that are heterozygous (meaning they have one copy of each allele).
The sum of these terms is always equal to 1, as it represents the entire population.
This equation assumes several conditions: that the population is large, randomly mating, with no mutations, migration, natural selection, or genetic drift.
In reality, these conditions are rarely met, and the Hardy-Weinberg equation serves as a useful model for understanding how allele frequencies can change over time due to various evolutionary influences.
For similar question on Hardy-Weinberg equation
https://brainly.com/question/29776155
#SPJ11
The sales tax in a certain city is 2%. A man buys a camera and pays a sales tax of $3.20. What was the cost of the camera, exclusive of the sales tax?
Answers
Divide the amount of tax by the rate of tax.
3.20 / 0.02 = 160
The cost of the camera is $160
If the sales tax in a certain city is 2% and man paid sales tax of $3.20 then the cost of the camera, exclusive of sales tax is $156.80.
What is sales tax?Sales tax is the tax that is paid by buyer to seller on the amount of goods sold.
How to find cost?We have been given that sales tax in a certain city is 2% and sales tax paid by man for camera is $3.20 and we have to find the cost of the product excluding sales tax.
Sales tax=tax percentage*total price
3.20=2/100*total price
Total price=3.20*100/2
=$160
Price excluding sales tax=total price - sales tax
=160-3.20
=157.80
Hence if the sales tax in a certain city is 2% and man paid sales tax of $3.20 then the cost of the camera, exclusive of sales tax is $156.80.
Learn more about sales tax at https://brainly.com/question/9437038
#SPJ2
10) The ratio of boys to girls in Mrs. Ronilo's math class is 3 to 1. What PERCENT of the class is girls
Answers
Answer:
25%
Step-by-step explanation:
ratio is 3B : 1G (this means, out of 4 sections, 3 are boys and 1 are girls)
this means that 3/4 of class is boys and 1/4 are girls
1/4 equals 25%
(-6,-4), (3,-4) find the slope
Answers
Answer:
Slope is 9.
Step-by-step explanation: The formula to find slope is m= y2-y1/x2-x1
-6=y1, -4=x1/ 3=y2, -4=x2
3-(-6)/-4-(-4)
=9/0=9
The Slope is 9.
I hope that this has helped, have a good day.
A function g models a relationship in which the dependent variable is multiplied by 9
for every 2 units the independent variable increases. The value of the function at 0 is 2.
Which equation represents g?
Answers
The link represented by function g multiplies the dependent variable by 9 for every increase of 2 units in the independent variable. The equation that represents g is \(g(x)= 2 \times3^x\),and the function has a value of 2 at 0
Let x be the independent variable, and let g(x) be the dependent variable.
According to the problem, for every 2 units the independent variable increases, the dependent variable is multiplied by 9. This suggests an exponential relationship of the form:
\(g(x)= a \times b^x\)
where a is the initial value of the dependent variable (when x=0), and b is the growth factor (how much the dependent variable changes for every unit increase in x).
We are given that g(0) = 2, so we can plug in these values and solve for a:
\(g(0)= a \times b^0\)
a = 2
So now we have:
\(g(x)= 2 \times b^x\)
We still need to find the growth factor b. We are told that the dependent variable is multiplied by 9 for every 2 units the independent variable increases. In other words:
\(g(x+2) = 9 \times g(x)\)
Using our equation for g(x), we can substitute in and simplify:
\(2 \times b(x+2) = 9 \times 2\times b^x\)
Simplifying, we get:
\(b^2\) = 9
Taking the square root of both sides (we take the positive square root since b must be positive in order for g to be an increasing function), we get:
b = 3
So the final equation for g(x) is:
\(g(x) = 2 \times 3^x\)
For more such questions on variable
https://brainly.com/question/25223322
#SPJ4
Determine whether the following statement is true or false. Explain.Inferences based on voluntary response samples are generally not reliable.A. False, because individuals who volunteer are least likely to have personal bias.B. True, because it is often the case that the individuals who volunteer do not accurately represent the population.C. True, because the group that volunteers may not be a large enough sample size.D. False, because a surveyor cannot force volunteers to respond.
Answers
The concept given is Sampling, which selects the group that you will actually collect data from in your research. The answer is option B, True; because it is often the case that the individuals who volunteer do not accurately represent the population.
Sampling is the selection of a subset of individualities from within a statistical population to estimate the characteristics of the whole population. Statisticians essay to collect samples that are representative of the population in question.
Frequently, voluntary response samples oversample people who have strong opinions and under-sample people who do not watch important about the content of the check. Therefore consequences from a voluntary response sample aren't as secure as conclusions grounded on an arbitrary sample of the entire population under consideration.
Volunteer slice doesn't induce a representative sample, so, thus, would not be the favored choice of experimenters who were keen to be suitable to generalize their data to the whole population.
To know more about Sampling,
brainly.com/question/28196409
#SPJ4
- A quadratic model for a certain stock on Wall Street is P=-3d2 + 50d, where d
represents the days of trading and P is the price per share.
a. What is the maximum price per share of the stock?
Help please I’m stuck
Answers
Answer:
208.333
Step-by-step explanation:
A quadratic model for a certain stock on Wall Street is given by :
\(P=-3d^2+50d\) ...(1)
(a) We need to find the maximum price per share of the stock. For maximum value, put :
\(\dfrac{dP}{dd}=0\)
\(\dfrac{d(-3d^2+50d)}{dd}=0\\\\-6d+50=0\\\\d=\dfrac{50}{6}\\\\d=8.34\)
price per share of the stock is maximum when the value of d = 8.34. Put the value of d in equation (1).
\(P=-3(8.34)^2+50(8.34)\\\\P=208.333\)
So, the maximum price per share of the stock is 208.333.