Suppose you kick a football and its movement can be modeled by a parabola. after 1 second its height is 15 feet above ground, after 2 seconds its height is 14 feet above ground, and after 3 seconds its height is 9 feet above ground.
a.find the equation of the parabola that models this behavior. y = -2x2 + 5x +12
b.after how many seconds does the ball hit the ground? 4 seconds
The equation of the parabola that models the movement of the football is y = -2x² + 5x + 12. The time at which the ball hits the ground is x = 4 seconds.
What is a quadratic equation?The quadratic equation is defined as a function containing the highest power of a variable is two.
a. We know that the general form of the quadratic equation is y = ax² + bx + c, where a, b, and c are constants.
We are given three points that the football passes through: (1, 15), (2, 14), and (3, 9).
We can substitute these points into the quadratic equation and solve for a, b, and c.
For the first point, we have: 15 = a(1)² + b(1) + c
For the second point, we have: 14 = a(2)² + b(2) + c
For the third point, we have: 9 = a(3)² + b(3) + c
Solving this system of equations using elimination gives us:
a = -2
b = 5
c = 12
Therefore, the equation of the parabola that models the movement of the football is y = -2x² + 5x + 12.
b. To find the time at which the ball hits the ground, we can set the value of y equal to 0 and solve for x.
y = -2x² + 5x + 12
0 = -2x² + 5x + 12
-12 = -2x² + 5x
-2x² + 5x - 12 = 0
We can use the quadratic formula to solve for x:
x = (-5 +/- √(5² - 4(-2)(-12)))/(2(-2))
x = (-5 +/- √(25 + 96))/(-4)
x = (-5 +/- √(121))/(-4)
x = (-5 +/- 11)/(-4)
Therefore, x = 4 or x = -3/2.
The time at which the ball hits the ground is x = 4 seconds.
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Would you use SSS or SAS to prove the triangles congruent? If there is not enough information to prove the triangles by SSS or SAS, write "not enough information". Explain your answer
Given that
∠A≅∠B, Gavin conjectured that ∠A and ∠B are complementary angles.
Which statement is a counterexample to Gavin 's conjecture?
m∠A=45° and m∠B=45°
m∠A=25° and m∠B=25°
m∠A=10°and m∠B=15°
m∠A=30°and m∠B=60°
Answer: It's B
B) m∠A=25° and m∠B=25°
Two students are to be selected at random from a class with 10 girls and 12 boys. what is the probability that both will be girls?
Mei is writing a coordinate proof involving a right isosceles triangle. Mei places her triangle on the coordinate plane such that one of the legs of the triangle lies along the x-axis and the other leg is parallel to the y-axis.
~ What coordinates should she assign to this third vertex of the right isosceles triangle?
Answer: (a,a)
Step-by-step explanation:
Given: A right isosceles triangle which is placed on the coordinate plane such that one of the legs of the triangle lies along the x-axis and the other leg is parallel to the y-axis.
From the given figure one vertex is on origin(0,0) and second on x -axis as (a,0) .ii.e. one side of the triangle is on the x axis with length 'a' and the side parallel to the y axis should perpendicular to the x axis with length 'a'.(∵ two sides are equal in a right isosceles triangle)
So the coordinates should she assign to the third vertex of the right isosceles triangle is (a,a).[as the point lie above the point (a,0), so its x abscissa should be same]
The third vertex of a right isosceles triangle where one leg lies on the x-axis and the second leg is parallel to the y-axis would have equal x and y coordinates, assuming the second vertex is at (a,0). Therefore, the third vertex would be at (a,a), for any real numerical value of 'a'. This makes use of key concepts in coordinate geometry.
Explanation:In the context of mathematics, specifically coordinate geometry, Mei can assign the third vertex of this right isosceles triangle to any point that is equally distant from both the x and y axes. Assuming the other two vertices of the triangle are at the origin (0,0) and on the x-axis (a,0), then the coordinates of the third vertex would be (a,a).
Right isosceles triangle is a special type of triangle where two sides are equal in length and the angle between them is 90 degrees. Since the legs of the triangle are along the x and y axes, this leads to the third vertex being on the line y=x.
A coordinate proof involves using the properties and definition of the geometric figures in the coordinate plane. It involves calculations with the coordinates of the vertices of the figures. In the case of coordinate geometry, it's essential to understand the placement of points on the x-y plane.
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Enter the solution to the inequality in the box.
7−3(x+5)+8x≤17
Answer:
The solution to the given inequality is (-∞,5].
Step-by-step explanation:
An inequality is a relation which makes a non-equal comparison between two numbers.
Given : 7-3(x+5)+8x≤17
=7-3(x+5)+8x ≤ 17
= 7- 3x - 15 + 8x ≤ 17
= -8 + 5x ≤ 17
Adding 8 on both sides:
= -8 +8+ 5x ≤ 17+8
= 5x ≤ 25
x ≤ 5
The solution to the inequality : (-∞,5].
What is another name for this ?
Solve for x. −2(x−3)−2x=18 Enter your answer in the box. x =
Suppose f is a linear function such that f(5) = 10 and f(9) = 3. find the equation for f.
The equation for the function f would be; f(x) = -1.75x + 18.75.
What is an equation?An equation is an expression that shows the relationship between two or more numbers and variables.
Function is a type of relation, or rule, that maps one input to specific output.
Given that f is a linear function such that f(5) = 10 and f(9) = 3, then
Consider that the function of x would be;
f(x) = -1.75x + 18.75
Now plug x = 5 in the given function, if it satisfy the function
f(5) = -1.75(5) + 18.75
f (5) =10
Again plug x = 9 in the given function, if it satisfy the function
f(9) = -1.75(9) + 18.75
f(9) =3
Therefore, the equation for the function f would be; f(x) = -1.75x + 18.75.
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Please help me , algebra math !!
Dinosaur fossils are often dated by using an element other than carbon, like potassium-40, that has a longer half life (in this case, approximately 1.25 billion years). suppose the minimum detectable amount is 0.1% and a dinosaur is dated with 40k to be 67 million years old. what is the maximum age of a fossil that we could date using 40k? (round your answer to one decimal place.)
The maximum age of a fossil that could be dated using 40K is approximately 5.36 billion years.
Explanation:Potassium-40 (40K) has a half-life of 1.25 billion years.
The minimum detectable amount for dating using 40K is 0.1%. If a dinosaur is dated using 40K to be 67 million years old, we can calculate the maximum age of a fossil that could be dated using 40K.
To find the maximum age, we can set up a proportion using the half-life of 40K:
(67 million years) / (1.25 billion years) = (x years) / (100%)
Solving for x, we get:
x = (67 million years) * (100%) / (1.25 billion years)
Calculating this, we find that the maximum age of a fossil that could be dated using 40K is approximately 5.36 billion years.
You buy 12 tickets and the total is $120. a student ticket is $9 and an adult ticket is $12. how many student tickets did you buy
Which property is used to simplify the following expression?
3(y-14)
A.
commutative property
B.
distributive property
C.
associative property
D.
inverse property
If f(x) = x2 + 7, what is the equation for f–1(x)?
To find the inverse of f(x) = x² + 7, replace f(x) with y and solve for x, resulting in the inverse function f–1(x) = √(x - 7), considering the domain x ≥ 7.
Explanation:To find the inverse function of f(x) = x² + 7, we must first replace f(x) with y, giving us y = x² + 7. To find the inverse, we solve for x. We start by subtracting 7 from both sides to get y - 7 = x². Taking the square root of both sides, we get x = ±√(y - 7).
Since the square root has two values (positive and negative), we choose the one that matches the domain and range of the original function. If f(x) is defined for all nonnegative values of x, the inverse would be f–1(x) = √(x - 7). If f(x) includes negative values of x, then we would consider both, resulting in two different functions.
However, because we're dealing with f(x) that implies only one output for each input without restrictions, we consider the principal square root. Thus, the equation for the inverse function is f–1(x) = √(x - 7), considering the domain where x is greater than or equal to 7.
Evaluate: −9^4
A) −36
B) −6,561
C) 36
D) 6,561
How many two-person conversations can occur at a party with 150 people?
---------------- conversations
Final answer:
There can be a total of 11,175 two-person conversations at the party.
Explanation:
In a party with 150 people, the number of two-person conversations that can occur can be calculated using the formula for combinations. Since each conversation involves two people, we need to choose 2 people from the total of 150. The formula for combinations is:
C(n, r) = n! / (r!(n-r)!)
Where n is the total number of people and r is the number of people we need to choose. Plugging in the values, we have:
C(150, 2) = 150! / (2!(150-2)!)
C(150, 2) = (150 ×149) / (2 ×1)
C(150, 2) = 11175
Therefore, there can be a total of 11,175 two-person conversations at the party.
Final answer:
At a party with 150 people, there can be 11175 unique two-person conversations, calculated using the combination formula 150C2.
Explanation:
The question is asking how many two-person conversations can occur at a party with 150 people. This is a classic combination problem where we need to calculate the number of unique pairs that can be made from 150 people. Since the order in which the pair is chosen does not matter, we use the combination formula nCr = n! / (r!(n-r)!), where 'n' is the total number of people and 'r' is the group size (in this case, 2 for a two-person conversation).
For 150 people, the calculation is:
150C2 = 150! / (2!(150-2)!) = 150! / (2! x 148!) = (150 x 149) / (2 x 1) = 11175.
Therefore, there can be 11175 unique two-person conversations at a party with 150 people.
M(3, 4) is the midpoint of mc010-1.jpg The coordinates of S are (4, 1). What are the coordinates of R?
Answer:
its c
Step-by-step explanation:
What is the equation for the line of reflection that maps the trapezoid onto itself? x = 0 x = 3 y = 3 y = 0?
The correct answer is option 3. The equation for the line of reflection that maps the trapezoid onto itself is y = 3.
To determine the line of reflection for a trapezoid that maps onto itself, we need to identify the axis of symmetry.
We can analyze each one:
1. x = 0 represents a vertical line of reflection. If the trapezoid were reflected over this line, the left and right sides would be swapped. This would not map the trapezoid onto itself unless the trapezoid is symmetric about the y-axis, which is not specified.
2. x = 3 also represents a vertical line of reflection. This would reflect the trapezoid over a vertical line three units to the right of the y-axis. Again, this would not map the trapezoid onto itself unless the trapezoid is symmetric about the line x = 3, which is not specified.
3. y = 3 represents a horizontal line of reflection three units above the x-axis. If the trapezoid has its bases parallel to the x-axis and the midline of the trapezoid is at y = 3, then reflecting over this line would indeed map the trapezoid onto itself. This is because the top half of the trapezoid would be a mirror image of the bottom half across the line y = 3.
4. y = 0 represents a horizontal line of reflection along the x-axis. Reflecting the trapezoid over the x-axis would flip it upside down, which would not map the trapezoid onto itself unless the trapezoid is symmetric about the x-axis, which is not typical for a trapezoid.
The complete question is:
What is the equation for the line of reflection that maps the trapezoid onto itself?
1. x = 0
2. x = 3
3. y = 3
4. y = 0
Alicia is writing the program for a video game. For one part of the game, she uses the rule (x,y) -> (x - 3, y + 4) to move points on the screen.
(a) What output does the rule give when the input is (-6,0)? Show your work.
(b) What output does the rule give when the input is (3,-4)? Show your work.
A) (-6,0) = -6-3, 0+4 so it moves to (-9,4)
B) (3,-4) = 3-3, -4+4 so it moves to (0,0)
Cost of a comic book: $3.95
Markup: 20%
suppose a population of 250 crickets doubles in size every six months. How many crickets will there be after two years?
double every 6 months
in 6 months there would be 250*2 = 500 crickets
in 1 year there would be 500*2 = 1000 crickets
in 1.5 years there would be 1000 *2 = 2000 crickets
in 2 years there would be 2000 *2 = 4000 crickets
Jason drives 400 miles in 10 hours. Which unit of measure is most appropriate for speed?
The sum of twice a number and half the number is 10 ???
Which expressions are equivalent to 3(62)? Select each correct answer. 3(60 + 2) 3(6 + 2) 3(6 + 56) 3(60 + 20)
A car travels 85 km from town a to town b, then 45 km from town b to town
c. the total trip took 1.5 h. what was the average speed of the car?
Final answer:
The average speed of the car is 86.67 km/h.
Explanation:
The average speed of a car can be calculated by dividing the total distance traveled by the total time taken. In this case, the car traveled 85 km from town A to town B and then 45 km from town B to town C, for a total distance of 85 km + 45 km = 130 km. The total trip took 1.5 hours. To find the average speed, divide the total distance by the total time: 130 km / 1.5 h = 86.67 km/h.
Find the polynomial equation of least degree with roots -1, 3, and (+/-)3i
The polynomial equation of least degree with the roots -1, 3, and (+/-)3i is x⁴ - 2x³ + 10x² + 2x - 27 = 0.
We have to find the polynomial equation of least degree with the given roots -1, 3, and (+/-)3i. As complex roots occur in conjugate pairs, if 3i is a root, then -3i must also be a root. Therefore, our polynomial will be of degree 4 since it has four roots in total.
The polynomial of least degree with roots x1, x2, x3, x4 can be expressed as:
(x - x1)(x - x2)(x - x3)(x - x4) = 0
Plugging in the roots given, we get:
(x + 1)(x - 3)(x - 3i)(x + 3i) = 0
To simplify this expression, let's first multiply the complex factors:
(x - 3i)(x + 3i) = x² - (3i)² = x² + 9
Now, multiply the remaining real factors with the result we just got:
(x + 1)(x - 3)(x² + 9) = 0
Expanding this, we obtain the polynomial equation:
x⁴ - 2x³ + 10x² + 2x - 27 = 0
This is the simplified equation of the polynomial with roots -1, 3, and (+/-)3i.
If f(x)=3x^2-2x+4 and g(x)=5x^2+6x-8 find (f+g)(x)
How many inches are there in 62 centimeters? there are 2.54 centimeters in 1 inch?
PLEASE HELP I NEED THIS ASAP!!!
Part A: Compare the data in the table with the relation f(x) = 3x – 10. Which relation has a greater value when x = 8? (2 points)
Part B: Using the relation in Part A, what is the value of x if f(x) = 80? (5 points)
THIS IS THE TABLE!!!
Input
(x) 2 4 6 8
Output
(y) 1 2 3 4
A scientist now has exactly 4.365 liters of water in a container. There were 5 liters of water in the container, but some of the water evaporated.How much water evaporated from the container?
Answer:
amount evaporated = 0.635 liters
Step-by-step explanation:
The scientist initially had 5 liters of water in a container . After evaporation took place he had only 4.365 liters of water remaining in the container.
The amount of water that evaporated can be computed when you remove the amount of water remaining after evaporation from the initial amount before evaporation.
Mathematically,
amount evaporated = initial amount before evaporation - amount remaining after evaporation
initial amount before evaporation = 5 liters
amount remaining after evaporation = 4.365 liters
amount evaporated = 5 - 4.365
amount evaporated = 0.635 liters