Two water balloons were launched into the air at different moments and collided. The water balloons were modeled by the quadratic functions: y = −3x2 + 24x + 3 and y = −4x2 + 25x + 15, where y represents the height in meters and x represents the time in seconds after the launch. What is the time, in seconds, that the balloons collided at the highest point?
Answer:
4s
Step-by-step explanation:
Equate the equations and solve for x:
[tex]-3x^2 + 24x + 3=-4x^2 + 25x + 15[/tex]
[tex]\implies x^2 -x -12=0[/tex]
[tex]\implies x^2 -4x+3x -12=0[/tex]
[tex]\implies x(x-4)+3(x-4)=0[/tex]
[tex]\implies (x+3)(x-4)=0[/tex]
[tex]\implies x=-3, x=4[/tex]
As time is positive, x = 4
The time at which the water balloons collided at their highest point is 7 seconds after the launch.
To find the time at which the two water balloons collided at their highest point, we need to determine the time at which the sum of their heights is maximized. The sum of the heights is represented by the sum of the two quadratic functions.
Given the functions:
y = -3x² + 24x + 3
y = -4x² + 25x + 15
The sum of the heights, S(x), is:
S(x) = (-3x² + 24x + 3) + (-4x² + 25x + 15)
Combine like terms:
S(x) = -7x² + 49x + 18
To find the time when the sum of the heights is maximized, we need to find the x-coordinate of the vertex of the parabolic function S(x). The x-coordinate of the vertex of a quadratic function in the form ax² + bx + c is given by:
x vertex = -b / (2a)
In this case, for S(x) = -7x² + 49x + 18:
a = -7
b = 49
Calculate the x-coordinate of the vertex:
x vertex = -49 / (2 * -7)
x vertex = 7
Now that we have the x-coordinate of the vertex, we can find the corresponding time by plugging it into either of the original functions. Let's use the first function for simplicity:
y = -3x² + 24x + 3
When x = 7:
y = -3(7)² + 24(7) + 3
y = -147 + 168 + 3
y = 24
Therefore, the time at which the water balloons collided at their highest point is 7 seconds after the launch.
To know more about quadratic functions:
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A triangle has angle measurements of 13°, 104°, and 63°. What kind of triangle is it?
Answer:
The triangle is a scalene obtuse triangle
Step-by-step explanation:
We have angles of 13, 104, and 63
Since all of the angles are different, all of the sides have different lengths, so the triangle is scalene
The largest angle is 104 degrees. Since it is larger than 90 degrees, it is obtuse.
The triangle is a scalene obtuse triangle
Answer:
obtuse maybe
Step-by-step explanation:
I don't really know but let me know if you get it right.
Why I picked obtuse is because of 104 degrees normally obtuse angles have smaller degrees than there is 1 that is bigger number for instantance 104 or over 100 degrees.
Have a nice day! :)
class of 28 13 boys and 15 girls
Answer: 15/13
Step-by-step explanation:
Given the functions F(x) = 6x + 11 and g(x) = × + 6, which of the following functions represents fla(x)] correctly?
F(x) = 6x + 11 and g(x) = x+6
f(g(x)) = 6(x+6) + 11 = 6x + 36 + 11 = 6x + 47
Hope that helps!
Noah bought a new car. It gets 21.7miles per gallon of gas on the highway. His gas tank is able to hold 14 gallons of gas. How many miles can Noah travel on the highway on one tank of gas?
Answer:
303.8
Step-by-step explanation:
All you do for this is 21.7 * 14
Help please i don’t know how to do it
Answer:
Step-by-step explanation:
[tex]\frac{dy}{dx}=x^{2}(y-1)\\\frac{1}{y-1} \text{ } dy=x^{2} \text{ } dx\\\int \frac{1}{y-1} \text{ } dy=\int x^{2} \text{ } dx\\\ln|y-1|=\frac{x^{3}}{3}+C\\[/tex]
From the initial condition,
[tex]\ln|3-1|=\frac{0^{3}}{3}+C\\\ln 2=C[/tex]
So we have that [tex]\ln |y-1|=\frac{x^{3}}{3}+\ln 2\\e^{\frac{x^{3}}{3}+\ln 2}=y-1\\2e^{\frac{x^{3}}{3}}=y-1\\y=2e^{\frac{x^{3}}{3}}+1[/tex]
Which of the following is the graph of the function y = 10 times 1.08 Superscript x?
a.
On a coordinate plane, a curve approaches the x-axis in quadrant 2, and then increases through (0, 10) to quadrant 1.
c.
On a coordinate plane, a curve approaches the x-axis in quadrant 3, and then decreases through (0, negative 10) to quadrant 4.
b.
On a coordinate plane, a curve approaches the x-axis in quadrant 2, and then increases through (0, 16) to quadrant 1.
d.
On a coordinate plane, a curve approaches the x-axis in quadrant 2, and then increases through (0, 5) to quadrant 1.
Answer:
a)
Step-by-step explanation:
The y-intercept is when x = 0.
[tex]a^0=1[/tex], therefore when x = 0, y = 10
So the y-intercept is (0, 10)
The curve approaches the x-axis in quadrant 2, increases, passes through (0, 10) to quadrant 1 (see attached graph)
ms. weaver bought 216 pencils for her class. The pencils came in pack of 24 how . many packs did she buy
Answer:
9 packs
Step-by-step explanation:
Step 1: Divide the number of pencils by the single pack that they came in.
216/24 = 9
Therefore, Ms. Weaver bought 9 packs for her class.
Verify: 9(24) = 216
The tenputurr is -2 C.If the the temperature rises by 15 c, what is the new temperature
Answer:
13 degrees Celsius
Step-by-step explanation:
15-2=13
What is the value of the exponent when simplifying the following expression: 16^1/2•16^1/2
Select one or more:
A. 1/4
B. 1/2+1/2
C. 1
D. 0
E. 1/2•1/2
F. 1/2-1/2
Answer:
B.
Step-by-step explanation:
because they have the same base and it's a multiplication so you have to add the exponents
simplifying the following expression: 16^1/2•16^1/2
C. 1
who can answer this have 20 points
First note that [tex]A_n[/tex] means the nth term in the sequence.
1. First one is a simple arithmetic progression/sequence as the common differences between each two numbers are 1
So the sequence will be
1.) 2,3,4,5,6,7...
And the rule since its +1 each time will be
[tex]A_n=2+1*(n-1)[/tex]
For the nth term of the sequence.
2.This one can be seen as a simple geometric sequence as the common ratio between the number and the number after it is 1:2
So the sequence will be
2.)2,4,8,16,32,64
And the rule since it is *2 each time is a simple exponential function
[tex]A_n=2*2^{n-1}\\or\\A_n=2^n[/tex]
3.This one is tricky but it can be seen that the term is double the term before subtract 1
So the sequence will be
3.)5,9,17,33,65,129
And the rule based on this sequence will be
[tex]A_n=A_{n-1}*2-1[/tex]
**Please note that this rule specifically is written in recursive form as it is a lot more difficult otherwise.
4 This one is similar to problem 1 which is an arithmetic sequence this one just have a common differences of 3 instead of 1.
So the sequence will be
4.)25,28,31,34,37
And the rule based on the sequence will be
[tex]A_n=25+3*(n-1)[/tex]
I hope this helps and if you don't understand any part of my explanation I would gladly help!
a. Use what you've learned in this unit to model the population of Western
Lowland Gorillas after 5, 10 and 20 years. Let y equal the population of the
gorillas and x represent the number of years since 2022. Show your work.
b. Use the information calculated in step A to create a table showing the Gorilla
population after 5, 10 and 20 years.
c. Explain why the table shows exponential decay. Summarize how scientists
can use exponential decay to predict population changes in endangered
species. Summarize your answer in 1-2 paragraphs.
An exponential function can either represent population growth or population decay
How to model the populationThe question is incomplete, as the resources to model the population of the Western Lowland Gorillas are not given.
However, the following is a general guide to solve the question.
An exponential function is represented as:
[tex]y=a(1 \pm r)^x[/tex]
Where:
(a) represent the initial value i.e. the initial population of the Western Lowland Gorillas (r) represents the rate at which the population increases or decreases.(x) represents the number of years since 2022(y) represents the population in x yearsGiven that the population of the Western Lowland Gorillas decreases, then the rate of the function would be 1 -r (i.e. an exponential decay)
Take for instance:
[tex]y = 20000 * 0.98^x[/tex]
By comparison.
a = 2000 and 1 - r = 0.98
The above function can be used to model the population of the Western Lowland Gorillas
Read more about exponential functions at:
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subtract: write each in simplest form. 1/2-1/4=
how do I do this problem in the simplest form.
1/2 - 1/4 =
1/2-1/4
Find its LCM which would be 4
So change the 1/2 to 2/4 and keep 1/4 the same since we already have the denominator we need so
2/4-1/4 = 1/4
The radius of a circle is 1 foot. What is the circle's area?
r=1 ft
Use 3.14 for .
Answer:
A=πr2=π·12≈3.14159ft²
A≈3.14ft²
Step-by-step explanation:
What the answer for this???
Use the provided hundredths model (and two colors) to demonstrate how to model your answer on #1.
Be sure to explain which cool represents each factor! Be sure to write the product!
Thanks for helping me, appreciate it alot!
Answer:
Send the question please. You can send in the comments or in a separate question.
Find the value of x. if 2xsquare+4=54 is given.
anybody please Hrd can u sole this pleaseeeeeeeeeeeee. soon ASAP...
1i) 3m. 3 x m is the same as 3m
1ii) 2m + 15. 2 x m + 15 can be written as 2m + 15
Answer:
3m and 2m+15
Step-by-step explanation:
"three times m" = 3 multiplied by m = 3m
"15 more than twice the given number"
Twice the given number is 2m. 15 more than that is 2m+15.
please help what is the answer and I will help me alot
Answer:
he will have $832.20 in his account if it is adding the rate if it is subtracting the rate then it is $627.80
Step-by-step explanation:
you first need to find out what is 2.8% of $730. this number ends up being $20.44 and you multiply it by 5 because the account exists for 5 years. then depending on if it is adding or subtracting to the account. if it is adding then it will be $832.20, and if it is subtracting it will be $627.80
P.S. This is all done by a kid in middle school.
Help me pleaseee will give out lots of points
Answer:
x=73
Step-by-step explanation:
The two angles are equivalent, therefore x=73
Brainliest if correct
Answer:
2x -12
Step-by-step explanation:
Multiply -3 times x and -3 times 4 to get -3x and -12. Then combine 5x -3x to get 2x - 12
Answer/Step-by-step explanation:
Distribute
-3(x+4)+5x
-3x-12+5x
Combine like terms
2x-12
[RevyBreeze]
Fill in the table with the next four terms of the sequence a1=4 and an=3an – 1, if n≥2.
Answer:
12, 36, 108, 324
Step-by-step explanation:
using the recursive formula and a₁ = 4 , then
a₂ = 3a₁ = 3 × 4 = 12
a₃ = 3a₂ = 3 × 12 = 36
a₄ = 3a₃ = 3 × 36 = 108
a₅ = 3a₄ = 3 × 108 = 324
Find the range of f(-5)=-2|x-5|+8 please with explination.
Answer:
Domain: set of possible input values (x-values)
Range: set of output values (y-values)
The range for the function [tex]f(x)=-2|x-5|+8[/tex] is:
[tex]f(x)\leq 8[/tex]
(5, 8) is the vertex of the line
Because |x - 5| is absolute (so always positive) and it is multiplied by -2,
-2|x - 5| will always be negative unless x = 5 (when it will be zero). Therefore, the max y-value is 8.
However, your question actually asked what the function is when x = -5:
[tex]f(-5)=-2|x-5|+8\\\\= -2|-5-5|+8\\\\= -12[/tex]
Step-by-step explanation:
Thats the all rhe answers step by step
Alia and Zarius are each reading the same book. Alia read 1/2 of the book on Monday. She read 1/3 of the book on Tuesday. Zarius read 4/8 of the book on
Tuesday and 1/4 on Monday. Zarius said he read more than Alia. Is he correct? Please explain why?
Answer:
I believe Alia because she read 2.5/3 of the book and Zarius only read 3/4 of the book altogether.
Step-by-step explanation:
Z; 4/8=1/2 or 2/4 +1/4=3/4
A: 1/2=3/6+2/6= 5/6 if you divide that by 2 then she read 2.5/3 of her book
in then end 3rds are greater then fourths
Liam by two boxes of tomatoes one box has two rows with eight tomatoes in each row the other box has eight rows with two tomatoes in each row
In a company's first year in operation, it made an annual profit of $242,500. The profit of the company increased at a constant 22% per year each year. How much total profit would the company make over the course of its first 30 years of operation, to the nearest whole number?
Answer:
total profit = $428,517,224 (nearest whole number)
Step-by-step explanation:
Use geometric sum formula:
[tex]S_n=\dfrac{a(1-r^n)}{1-r}[/tex]
where [tex]a[/tex] is the initial value and [tex]r[/tex] is the common ratio
We have been told that the initial value is 242500, so [tex]a=242500[/tex].
If the company's profit increases by 22% per year, this means each year's profit is 122% of the previous year's profit. 122% = 122/100 = 1.22
Therefore, [tex]r = 1.22[/tex]
We need to calculate the total profit earned over 30 years, so [tex]n = 30[/tex]
Inputting these values into the formula:
[tex]\implies S_{30}=\dfrac{242500(1-1.22^{30})}{1-1.22}=\$428,517,224 \ \textsf{(nearest whole number)}[/tex]
how do u disapprove a prof statement
WILL GIVE 100 POINTS AND BRAINLIEST
I need this answer IMMEDIATELY!!!
The graph models how a farmer's profit from a strawberry crop depends on the number of days the farmer waits after the first strawberries ripen before harvesting the crop.
Select from the drop-down menu to interpret the increasing interval of the graph.
The profit increases from
Choose...
days after the first strawberries ripen.
Answer:
The profit increases from day 0 to day 8 after the strawberries first ripen.
Step-by-step explanation:
A function is increasing when the slope is positive (y gets bigger as x gets bigger).
A function is decreasing when the slope is negative (y gets smaller as x gets bigger).
From inspection of the graph, we can see that the maximum point is (8, 6000), so the maximum profit of $6,000 occurs on day 8.
The slope of the curve is increasing as the curve approaches the maximum point, so the profit is increasing during this time period.
Therefore, the profit increases from day 0 to day 8 after the strawberries first ripen.
The slope of the curve is decreasing after the maximum point, so the profit is decreasing after 8 days.
The profit increases from 8 days after the first strawberries ripen.
When we observe the graph we see that the function reaches its maximum at y-axis on the point (8,6000) 6000 is the maximum profit, which is made after 8 days the strawberries have ripen.After 8 days the prices have started to decrease.Mrs. Douglas packs candy bars into boxes for the members of the school band to sell as a fundraiser. Each candy bar weighs 5.75 ounces, and the empty box weighs 2.95 ounces. If the algebraic rule y = 5.75x + 2.95 represents the total weight of the packed box, what does the variable x represent?
a. Weight of each candy bar inside a box
b. Total weight of a box filled with candy bars
c. Total number of boxes filled with candy bars
d. Number of candy bars packed into a box
Answer:
d
Step-by-step explanation:
If 9x + 1 = 1, find the value of x.
Answer:
the value of x is 0
Step-by-step explanation:
you: thanks
Me: no problem :)
What operation would isolate the variable -11 + m = 26
Answer:
the addition property
Step-by-step explanation:
to isolate the m you would add 11 to each side to cancel out -11 so m=37
