The depth of the water at the end of a pier changes periodically along with the movement of tides. On a particular day, low tides occur at 12:00 am and 12:30 pm, with a depth of 2.5 m, while high tides occur at 6:15 am and 6:45 pm, with a depth of 5.5 m. Let t = 0 be 12:00 am. Which periodic function, sine or cosine, would be a simpler model for the situation? Explain.
Answer: A cosine function would be a simpler model for the situation.
The minimum depth (low tide) occurs at
t = 0. A reflection of the cosine curve also has a minimum at t = 0.
A sine model would require a phase shift, while a cosine model does not.
Step-by-step explanation:
Using a cosine function is simpler to model the tide changes because it starts at the maximum value, aligning with the high tide occurrence.
The depth of the water at the end of a pier changes periodically due to tides. Given that low tides occur at 12:00 am and 12:30 pm with a depth of 2.5 m, and high tides occur at 6:15 am and 6:45 pm with a depth of 5.5 m, we need to model this situation with a periodic function.
Let's align this with a cosine function for simplicity. In general, the cosine function can be modeled as y = A cos(B(t - C)) + D, where:
A is the amplitude (half of the difference between high tide and low tide depth, (5.5 m - 2.5 m)/2 = 1.5 m)B determines the period (a full tide cycle is roughly 12 hours and 25 minutes or 747.5 minutes; B = 2π/747.5)C is the horizontal shift (shift corresponding to the time of the first high tide, 6.25 hours or 375 minutes, so C = 375)D is the midline of the function (average depth, (5.5 m + 2.5 m)/2 = 4 m)Therefore, the function becomes: y(t) = 1.5 cos((2π/747.5)(t - 375)) + 4. Using a cosine function is simpler because it starts at the maximum value, which corresponds to the high tide.
Use synthetic division to find P(–2) for P(x) = x4 + 9x3 - 9x + 2 .
A. –2
B. –36
C. 0
D. 68
-36 is the correct answer
find an ordered pair that is a solution to the equation x-4y=4
Ordered pairs that are solutions to the equation x-4y=4 is (4, 0) and (0, -1).
The given equation is x-4y=4.
We need to find an ordered pair that is a solution to the equation.
How to find the solution to an equation?The solutions of linear equations are the points at which the lines or planes representing the linear equations intersect or meet each other. A solution set of a system of linear equations is the set of values to the variables of all possible solutions.
From the graph, we can observe that (4, 0) and (0, -1) are solutions.
Verification of the solution (4, 0):
4-4y=4
⇒-4y=0
⇒y=0
Verification of the solution (0, -1):
0-4y=4
⇒-4y=4
⇒y=-1
Therefore, ordered pairs that are solutions to the equation x-4y=4 is (4, 0) and (0, -1).
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A bicycle store costs $2400 per month to operate. The store pays an average of $60 per bike. The average selling price of each bicycle is $120. How many bicycles must the store sell each month to break even? Write a system of equations to represent the situation, then solve.
Using linear functions, it is found that the store must sell 40 bicycles each month to break even.
What is a linear function?A linear function is modeled by:
y = mx + b
In which:
m is the slope, which is the rate of change, that is, by how much y changes when x changes by 1.b is the y-intercept, which is the value of y when x = 0, and can also be interpreted as the initial value of the function.A bicycle store costs $2400 per month to operate, and pays an average of $60 per bike, hence the cost function is given by:
C(x) = 2400 + 60x
The average selling price of each bicycle is $120, hence the revenue function is given by:
R(x) = 120x
It breaks even when cost equals revenue, hence:
R(x) = C(x)
120x = 2400 + 60x
60x = 2400
x = 240/6
x = 40.
The store must sell 40 bicycles each month to break even.
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To determine the number of bicycles the store must sell each month to break even, we can set up a system of equations. The store must sell 40 bicycles each month to break even.
Explanation:To determine the number of bicycles the store must sell each month to break even, we can set up a system of equations.
Let's say the number of bicycles sold per month is x.
The monthly operating cost of the store is $2400.
The cost of producing each bike is $60, so the total cost to produce x bikes would be $60x.
The average selling price of each bike is $120, so the total revenue from selling x bikes would be $120x.
To break even, the total revenue should equal the total cost, so we can set up the equation:
$120x = $60x + $2400
Simplifying the equation, we get:
$60x = $2400
Dividing both sides by $60, we find:
x = 40
Therefore, the store must sell 40 bicycles each month to break even.
Find H to the nearest degree.
Given the information below, find the coordinates of the vertices L and P such that ABCD=NLPM. A(2,0), B(2,4,), C(-2,4), D(-2,0), M(4,0), N(12,0)
Answer:
Its B I just took the test
A bakery sold apple pies for $11 and blueberry pies for $13. One Saturday they sold a total of 38 pies and collected a total of $460. How many apple pies did they sell and how many blueberry pies did they sell?
A= apple pie
B = blueberry pie
a+b=38
a=38-b
11a + 13b =460
11(38-b) + 13b = 460
418-11b +13b = 460
2b=42
b=42/2 =21
they sold 21 blueberry pies and 17 apple pies
Similar question to the last, just a bit more difficult
-8x-10y=24
6x+5y=2
solve for (x.y)
no clue on this one though
Which ordered pair is a solution to the system of inequalities? y <3 and y >-x+5 ?
A. (6,1)
B. (2,1)
C. (3,0)
D. (-2,4)
PS. for the equation y>-x+5, the sign should be greater than or equal to. I just can't find the key on my phone (>)
What number should be added to both sides of the equation to complete the square? x^2 â 10x = 7?
Simplify fraction 23076923076923/10000000000000
Find the value tan 39 degrees. Round to the nearest ten-thousandth
A ) 0.8098
B ) 0.6293
C ) 0.7771
D ) 3.6146
Dominique is thinking about buying a house. The table below shows the projected value of two different houses for three years.
House 1 (value in dollars) year 1: 286,000 year 2: 294,580 year 3: 303,417.40 House 2 (value in dollars) year 1: 286,000 year 2: 295,000 year 3: 304,000
Part A: What type of function, linear or exponential, can be used to describe the value of each of the houses after a fixed number of years? Explain your answer.
Part B: Write one function for each house to describe the value of the house f(x), in dollars, after x years.
Part C: Dominique wants to purchase a house that would have the greatest value in 25 years. Will there be any significant difference in the value of either house after 25 years? Explain your answer, and show the value of each house after 25 years.
The functions for House 1 and House 2 are [tex]\(f_1(x) = 5079.6x + 248900.4\)[/tex] and [tex]\(f_2(x) = 4666.\overline{66}x + 256000\)[/tex] respectively.
After 45 years, House 1 will be valued at $477,555, and House 2 will be valued at $465,000.
Here's the step-by-step solution with complete calculations for the given problem:
Part A: Identifying the Function Type
- House 1:
- Year 1 to Year 2: [tex]\(259,059.60 - 253,980 = 5,079.60\)[/tex]
- Year 2 to Year 3: [tex]\(264,240.79 - 259,059.60 = 5,181.19\)[/tex]
- House 2:
- Year 1 to Year 2: [tex]\(263,000 - 256,000 = 7,000\)[/tex]
- Year 2 to Year 3: [tex]\(270,000 - 263,000 = 7,000\)[/tex]
The functions are linear because the changes are constant for House 2 and almost constant for House 1.
Part B: Formulating the Equations
- House 1 Equation:
- Slope (m): [tex]\(5079.6\)[/tex]
- Y-intercept (c): [tex]\(248900.4\)[/tex]
- Equation: [tex]\(f_1(x) = 5079.6x + 248900.4\)[/tex]
- House 2 Equation:
- Slope (m): [tex]\(4666.\overline{66}\)[/tex]
- Y-intercept (c): [tex]\(256000\)[/tex]
- Equation: [tex]\(f_2(x) = 4666.\overline{66}x + 256000\)[/tex]
Part C: Calculating Future Values
- House 1 Value at Year 45:
[tex]- \(f_1(45) = 5079.6 \times 45 + 248900.4 = 477555\)[/tex]
- House 2 Value at Year 45:
[tex]- \(f_2(45) = 4666.\overline{66} \times 45 + 256000 = 465000\)[/tex]
These equations predict the values of the houses after 45 years based on the given data. House 1 will be valued at $477,555, and House 2 will be valued at $465,000.
Write a variable expression for 9 more than a number s.
The path of a ping pong ball that is hit from one end of the table can be modeled by the equation (y= -1/4 x^2 5x) where x is measured in inches and represents the horizontal distance from the edge of the table, and y represents the height of the ping pong ball in inches above the table. What is the maximum height of the ping pong ball?
A.
The ball reaches a maximum height of 30 inches above the table.
B.
The ball reaches a maximum height of 20 inches above the table.
C.
The ball reaches a maximum height of 25 inches above the table.
D.
The ball reaches a maximum height of 27 inches above the table.
What is the area of the obtuse triangle given below?
Answer:
Option (d) is correct.
The area of triangle is 38.5 square units
Step-by-step explanation:
Given: An obtuse triangle.
We have to find the area of this obtuse angle.
Consider the given obtuse triangle
Area of triangle = [tex]\frac{1}{2}\cdot b \cdot h[/tex]
where b = Base
h = height
Given : base = 11 units
and height = 7 units
Thus, Area of triangle = [tex]\frac{1}{2}\cdot 11 \cdot 7[/tex]
Simplify, we have,
Area of triangle = 38.5
Thus, The area of triangle is 38.5 square units
Which of the following is an extraneous solution of sqrt(-3x-2)=x+2 a.-6 b.-1 c.1 d.6
Answer:
- 6 is the extraneous solution.
Step-by-step explanation:
Given : [tex]\sqrt{-3x -2} = x + 2[/tex].
To find : Which of the following is an extraneous solution .
Solution : We have given that [tex]\sqrt{-3x -2} = x + 2[/tex].
Taking square both sides
-3x - 2 = [tex](x+2)^{2}[/tex].
On applying identity [tex](a+b)^{2}[/tex] = a² + b² + 2ab
Then ,
-3x -2 = x² + 2² + 2 * 2 *x
-3x -2 = x² + 4 + 4x.
On adding both sides by 3x
-2 = x² + 4 + 4x + 3x
-2 = x² + 4 + 7x
On adding both sides by 2
0 = x² + 4 + 7x + 2
On switching sides
x² +7x + 6 = 0
On Factoring
x² +6x + x + 6 = 0
x ( x+ 6 ) +1 (x +6 ) = 0
On grouping
( x +1) ( x +6) = 0
x = -1, -6.
Let check for x = -6
[tex]\sqrt{-3 (-6) -2} = -6 + 2[/tex].
4 = -4
An extraneous solution is a root of a transformed equation that is not a root of the original equation.
Therefore, -6 is the extraneous solution.
What is the distance between the points (22, 27) and (2, -10)? if necessary, round your answer to two decimal places.a. 57 units?
Answer:
42.06
Step-by-step explanation:
Two points (22,27) and (2,-10)
using distance formula:
[tex]d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}[/tex]
So, distance between (22,27) and (2,-10) is D
[tex]D=\sqrt{(22-2)^2+(27-(-10))^2}[/tex]
[tex]D=\sqrt{(20)^2+(37)^2}[/tex]
[tex]D=\sqrt{400+1369}[/tex]
[tex]D=\sqrt{1769}[/tex]
[tex]D=42.059[/tex]
Round off two decimal place.
[tex]D=42.06[/tex]
Hence, The distance between the points is 42.06
xy is displayed by a scale factor of 1.3 with the origin as the center of dialation to create the image xy. if the slope and length of xy are m what is the slope of xy
Answer:
he XY is dilated by a scale factor of 1.3 with the origin as the center of dilation to create the X'Y'. So the length of X'Y' is 1.3 times of origin but the slope is the same. The slope is m
Step-by-step explanation:
Best explained and correct answer gets brainliest.
total = 2372* (1+0.045)^20=
2372*1.045^20=
2372 * 2.411714025= 5720.585
she will have 5720.59 in 20 years
Is the relation {(3, 5), (–4, 5), (–5, 0), (1, 1), (4, 0)} a function? Explain. Type your answer below
{(3, 5), (–4, 5), (–5, 0), (1, 1), (4, 0)}
As long as there are the same x-value does not have multiple y-value results, it will be a function. This data array doesn't contain any recurring x-values. Therefore, this is a function (in simple terms of speaking).
Jerome bought 15 videos from a department store. Some videos were new releases, x, which cost $19, and some videos were classics, y, which cost $8. He spent a total of $164 on the videos. Which system of equations is set up correctly to model this information?
Answer:
The system of equations is
[tex]x+y=15[/tex]
[tex]19x+8y=164[/tex]
Step-by-step explanation:
Let
x------> the number of videos of new releases
y-----> the number of classics videos
we know that
[tex]x+y=15[/tex] ------> equation A
[tex]19x+8y=164[/tex] ------> equation B
Using a graphing tool
Solve the system of equations
Remember that the solution of the system of equations is the intersection point both graphs
The intersection point is [tex](4,11)[/tex]
see the attached figure
therefore
the number of videos of new releases is [tex]4[/tex]
the number of classics videos is [tex]11[/tex]
Answer:
Jerome bought 15 videos from a department store. Some videos were new releases, x, which cost $19, and some videos were classics, y, which cost $8. He spent a total of $164 on the videos. Which system of equations is set up correctly to model this information?
x + y = 15. 19 x + 8 y = 164.
x + y = 15. 8 x + 19 y = 164.
x + y = 164. 19 x + 8 y = 15.
x + y = 15. 19 x minus 8 y = 164.
ANSWER IS A
Solve x 2 + 9x + 8 = 0 by completing the square. What are the solutions?
x^2+9x+8=0
(x+1)(x+8)=0
x+1=0
x=-1
x+8 = 0
x=-8
solutions are -1 and -8
If the factors of a polynomial are x-2 and x-5, what values of x make that polynomial 0?
A. 1 and 2
B. -2 and -5
C. 2 and 5
D. Cannot be determined
The bulldog soccer team wants to increase the size of its
Practice field by a scale factor of 1.5. The field is a rectangle that currently measures 30 ft by 80 ft. The measurements of the new practice field should be 45 ft by ft.
multiply 80 by 1.5
80*1.5 = 120 feet
!!! HELP What values for (picture of equation) satisfy the equation?
Question 1- Heather wanted to find the density of a solution with a mass of 2.234
grams and a volume of 2.131 milliliters. She uses the density formula,
density = mass/volume.
If both her mass and volume were accurately measured to the thousandths place what is an accurate value for the density measured in g/mL?
Question 2- Coleton measures the sides of a rectangular piece of plywood. One
side is 72.6 inches long, and the shorter side is 36 inches long. What is the area of
the piece of plywood, rounded appropriately using significant figures?
Question 3- When would you want to use the median over the mean for
describing the measure of center for a data set?
identify the maximum and minimum values of the function y = 3 cos x in the interval [-2π, 2π]. Use your understanding of transformations, not your graphing calculator.
The maximum value of the function y = 3 cos x will be 3.
The minimum value of the function y = 3 cos x will be - 3.
What is an expression?Mathematical expression is defined as the collection of the numbers variables and functions by using operations like addition, subtraction, multiplication, and division.
Given that;
The function is,
⇒ y = 3 cos x
In the interval [-2π, 2π].
Now,
Since, The function is,
⇒ y = 3 cos x
Hence, We get;
The maximum value of the function y = 3 cos x is,
⇒ y = 3 cos2π
⇒ y = 3 × 1
⇒ y = 3
And, The minimum value of the function y = 3 cos x is,
⇒ y = 3 cos(-2π)
⇒ y = 3 × - 1
⇒ y = - 3
Thus, The maximum value of the function = 3.
The minimum value of the function = - 3.
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Please Help. Thank you
(a) a pizza parlor has a choice of 11 toppings for its pizzas. from these 11 toppings, how many different 7 -topping pizzas are possible?
To solve for the number of different possible pizzas with 7 toppings out of 11 and the arrangement of these toppings is not important, therefore we use the combination formula.
The formula for combination is:
n C r = n! / r! (n – r)!
where,
n = is the total number of toppings = 11
r = the number of toppings in a pizza = 7
Substituting the values into the equation:
11 C 7 = 11! / 7! (11 – 7)!
11 C 7 = 11! / 7! * 4!
11 C 7 = 330
Therefore there are a total different 330 pizzas with 7 different combinations toppings.
Final answer:
330 different combinations.
Explanation:
The student's question is about combinations in Mathematics, specifically how many 7-topping pizzas can be made from a choice of 11 toppings. This is a combinatorics problem, and the solution involves the use of the combination formula, which is given as:
C(n, k) = n! / (k!(n-k)!)
where n is the total number of items to choose from, k is the number of items to choose, n! denotes the factorial of n, and C(n, k) represents the number of combinations.
In this case, n is 11 (the total number of toppings) and k is 7 (the number of toppings on the pizza).
Therefore, the number of different 7-topping pizzas possible is:
C(11, 7) = 11! / (7!(11-7)!) = 11! / (7!4!) = (11x10x9x8)/(4x3x2x1) = 330
Hence, 330 different 7-topping pizzas are possible.