A spring is oscillating so that its length is a sinusoidal function of time. Its length varies from a minimum of 10 cm to a maximum of 14 cm. At t=0 seconds, the length of the spring was 12 cm, and it was decreasing in length. It then reached a minimum length at time t= 1.2 seconds. Between time t=0 and t=8 seconds, how much of the time was the spring longer than 13.5 cm?
Suppose S and T are mutually exclusive events. Find P(S or T) if P(S) = 65% and P(T) = 7%.
a. 4.55%
b. 72%
c. 58%
d. 455%
Answer: P(S\cup T)=72%
Step-by-step explanation:
We are given that S and T are mutually exclusive events.
Therefore, the intersection of both the events must be 0.
i.e. [tex]p(S\cap T)=0[/tex]
P (S) = 65% and P(T) = 7%
We know that P(S or T)=[tex]P(S\cup T)=P(S)+P(T)-P(S\cap T)[/tex]
[tex]\Rightarrow P(S\cup T)=65\%+7\%=72\%[/tex]
Hence, P(S\cup T)=72%
State whether each situation involves a combination or a permutation.
4 of the 20 radio contest winners selected to try for the grand prize
5 friends waiting in line at the movies
6 students selected at random to attend a presentation
a) permutation, combination, permutation
b) combination, permutation, permutation
c) combination, permutation, combination
d) permutation, combination, combination
Answer:
4 of the 20 radio contest winners selected to try for the grand prize : C
5 friends waiting in line at the movies: C
6 students selected at random to attend a presentation: P
Final answer:
The scenarios illustrate combination when order does not matter (selecting contest winners and student attendees) and permutation when order matters (friends in line). The correct sequence is combination, permutation, combination.
Explanation:
In the context of the scenarios provided, we need to differentiate whether the situations are examples of combinations or permutations. A permutation is an arrangement of objects where order matters, while a combination is a selection of objects where order does not matter.
4 of the 20 radio contest winners selected to try for the grand prize - This is a combination, as the order in which the winners are selected is not relevant.5 friends waiting in line at the movies - This is a permutation, as the order in which the friends are lined up matters.6 students selected at random to attend a presentation - This is a combination, as the order of selection does not impact which students attend.Thus, the correct answer to the sequence of scenarios is: combination, permutation, combination, which correlates with option c.
2s + 5 greater than or equal to 49
The value of s is greater or equal to 22.
What is inequality?It shows a relationship between two numbers or two expressions.
There are commonly used four inequalities:
Less than = <
Greater than = >
Less than and equal = ≤
Greater than and equal = ≥
We have,
2s + 5 greater than or equal to 49.
This can be written as,
(2s + 5) ≥ 49
Solve for s.
2s + 5 ≥ 49
2s ≥ 49 - 5
2s ≥ 44
s ≥ 22
Thus,
s is greater than or equal to 22.
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Write an algebraic expression which represents the volume of a box whose width is 4y, height is 6y and length is 3y + 1.
"if a snowball melts so that its surface area decreases at a rate of 1 cm 2 min, find the rate at which the diameter decreases when the diameter is 10 cm." (stewart 249) stewart, james. single variable calculus, 8th edition. cengage learning, 20150101. vitalbook file.
The sum of two consecutive terms in the arithmetic sequence 3, 6, 9, 12, ... is 303; find these two terms.
The first consecutive term of the arithmetic sequence is ?
The second consecutive term of the arithmetic sequence is ?
Final answer:
The first consecutive term of the arithmetic sequence that sums to 303 is 150. The second consecutive term is 153. We found this by setting up an equation for the sum of two consecutive terms and solving for the first term.
Explanation:
To find the two consecutive terms in the arithmetic sequence 3, 6, 9, 12, ... that sum up to 303, we first need to understand the properties of an arithmetic sequence. The given sequence has a common difference of 3 (that is, each term is 3 more than the previous term). Let's denote the first of these two consecutive terms as a. Therefore, the next term would be a + 3 (since the common difference is 3).
We are given that the sum of these two terms is 303, so we can write an equation:
a + (a + 3) = 303
Combining like terms, we get:
2a + 3 = 303
Subtracting 3 from both sides gives:
2a = 300
Dividing both sides by 2 gives:
a = 150
So, the first term is 150 and the second term, being a + 3, is 153.
Mrs. Jackson has $7,000 to invest. If she invests part at 6% simple annual interest and part at 8% simple annual interest, she will get an annual return of $520. How much should she invest at 8%?
If thewronskian w of f and g is 3e4t,and if f(t) = e2t,find g(t).
How much interest is gained if $250 is deposited in your bank account at the end of the year for each of the next 7 years? savings account pays 8% compounded annually?
John’s gross pay for the week is $500. He pays 1.45 percent in Medicare tax, 6.2 percent in Social Security tax, 2 percent in state tax, 20 percent in federal income tax, and $20 as an insurance deduction. He does not have any voluntary deductions. What is John’s net pay for the week?
John's net pay is calculated by subtracting deductions for Medicare, Social Security, state and federal taxes, and insurance from his gross pay of $500. The total deductions amount to $168.25, resulting in a net pay of $331.75.
Explanation:Calculation of John's Net Pay
To calculate John's net pay, we need to subtract all the deductions from his gross pay. Since his gross pay is $500, we will apply the following deductions:
Medicare tax: 1.45% of $500 = $7.25
Social Security tax: 6.2% of $500 = $31.00
State tax: 2% of $500 = $10.00
Federal income tax: 20% of $500 = $100.00
Insurance deduction: $20.00
Add up all deductions: $7.25 (Medicare) + $31.00 (Social Security) + $10.00 (State Tax) + $100.00 (Federal Tax) + $20.00 (Insurance) = $168.25
John's net pay is therefore calculated by subtracting the total deductions from his gross pay: $500.00 - $168.25 = $331.75.
The Sugar Sweet Company is going to transport its sugar to market. It will cost $5250 to rent trucks, and it will cost an additional $175 for each ton of sugar transported. Let C represent the total cost (in dollars), and let S represent the amount of sugar (in tons) transported. Write an equation relating C to S, and then graph your equation using the axes below.
The linear equation formed is C = 5250 + 175S, where C is the total cost and S is the amount of sugar in tons. This expresses the cost for Sugar Sweet Company to transport its sugar to the market, beginning from a fixed cost of $5250 with an additional $175 charged per ton of sugar transported.
Explanation:The question involves the creation of a linear equation that represents the total cost, C, of transporting sugar. We are told the initial cost of renting trucks is $5250 and there's an additional cost of $175 for each ton of sugar, S.
Therefore, we can write the equation as: C = 5250 + 175S.
To graph this equation, start at the point (0, 5250) on the y-axis which represents the initial cost. The slope of the line is 175, which means for each ton of sugar transported, the cost increases by $175. From the starting point, you can plot other points moving up vertically 175 units for each unit moved to the right horizontally.
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How many 2n-digit positive integers can be formed if the digits in odd positions (counting the rightmost digit at position 1) must be odd and the digits in even positions must be even and positive?
To find the number of 2n-digit integers where odd position digits are odd, and even position digits are even and positive, we calculate based on the choices for each position, giving us a result of (5^n) * (4^n).
Explanation:We encounter a combinatory problem in working out how many 2n-digit positive integers can be formed if the digits in odd positions must be odd and the digits in even positions must be even. Before proceeding, it's important to grasp the concept of positional numbering, where the rightmost digit is considered at position 1, and the counting proceeds from right to left.
For a 2n-digit positive integer, i.e., an integer with an even number of digits, there will be n digits at odd positions and n digits at even positions. For the odd positions, the digits can be any one of the five odd integers (1, 3, 5, 7, 9) and for the even positions, the digits can be any one of the four even positive integers (2, 4, 6, 8) because 0 is excluded as the question mentions they should be positive.
Therefore, for each position, we have a choice of five odd integers or four even integers. Since there are n odd positions and n even positions, we end up with (5^n) * (4^n) total possibilities or combinations.
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The total number of valid 2n-digit positive integers, where digits in odd positions must be odd and digits in even positions must be even, is calculated using the formula 5ⁿ * 4ⁿ.
To solve this problem, we need to consider the constraints given: digits in odd positions must be odd and digits in even positions must be even. Let's break down the problem step-by-step:
We have 2n-digit numbers. Therefore, there are n odd positions and n even positions.For the odd positions (1st, 3rd, 5th, ....., 2n-1), each digit can be 1, 3, 5, 7, or 9. So, there are 5 choices for each position.For the even positions (2nd, 4th, 6th, ....., 2n), each digit can be 2, 4, 6, or 8. There are 4 choices for each position.To find the total number of such 2n-digit positive integers, multiply the number of choices for all positions:Total combinations = (Number of choices for odd positions) n * (Number of choices for even positions) n = 5ⁿ * 4ⁿ
This is the formula to calculate the number of valid 2n-digit integers under the given constraints.
Rationalize the denominator of square root of negative 16 over open parentheses 1 plus i close parentheses plus open parentheses 6 plus 3 i.
A stocker put 57 boxes of detergent on the shelves in 2 minutes. After 5 minutes, he had put 117 boxes on the shelves. How many boxes were on the shelves when he started?
What is the next value.
4 D 7 G 10 J 13
The next value in the sequence is 16, following an increment of 3 in each step.
The next value in the sequence is 16.
The sequence increments by 3 starting from 4 (4, 7, 10, 13, ...)
Therefore, the next value after 13 would be 13 + 3 = 16.
What is the product of 119 thousandths times 10?
a. 119 hundreths
b. 119 thousands
c. 119 tenths
Answer:
119 hundreths
Step-by-step explanation:
To Find: What is the product of 119 thousandths times 10?
Solution:
Thousandths can be represented in the fraction form as[tex]\frac{1}{1000}[/tex]
So, 119 thousandths = [tex]\frac{119}{1000}[/tex]
Now to find the product of 119 thousandths times 10
[tex]\Rightarrow \frac{119}{1000} \times 10[/tex]
[tex]\Rightarrow \frac{119}{100} [/tex]
Now hundreths can be represented in the fraction form as[tex]\frac{1}{100}[/tex]
So, [tex]\Rightarrow \frac{119}{100} [/tex] = 119 hundreths.
Hence the product of 119 thousandths times 10 is 119 hundreths.
Thus Option A is True.
What is the value of 3×5+6×2+1
what is the gcf of 120 and 72
what is 1/3m-1-1/2n when m=21 and n=12
Answer:
Value of the expression is 0
Step-by-step explanation:
[tex]\frac{1}{3} m -1-\frac{1}{2} n[/tex]
Given the value of m and n
m= 21 and n= 12
We plug in the value of m and n in the given expression
[tex]\frac{1}{3} m -1-\frac{1}{2} n[/tex]
[tex]\frac{1}{3}(21) -1-\frac{1}{2}(12)[/tex]
[tex]\frac{21}{3} -1-\frac{12}{2}[/tex]
[tex]7-1-6= 0[/tex]
So the value of given expression is 0 when we plug in the values of m and n
If a boatman rows his boat 35km up stream and 55km downstream in 12 hours and he can row 30km upstream and 44 km downstream in 10hr , then the speed of the stream and that of the boat in still water
To answer this item, we let x be the speed of the boat in still water. The speed of the current, we represent as y.
When the boat travels upstream or against the current, the speed is equal to x – y and x + y if it travels downstream or along with the current.
The time it takes for the an object to travel a certain distance is calculated by dividing the distance by the speed.
First Travel: 35 / (x – y) + 55 / (x + y) = 12
Second travel: 30 / (x – y) + 44 / (x + y) = 10
Let us multiply the two equations with the (x-y)(x+y)
This will give us,
35(x + y) + 55(x – y) = 12(x-y)(x+y)
30(x + y) + 44(x – y) = 10(x-y)(x+y)
Using dummy variables:
Let a = x + y and b be x – y
35a + 55b = 12ab
30a + 44b = 10ab
From the first equation,
b = 35a/(12a – 55)
Substituting to the second equation,
30a + 44(35a/(12a – 55)) = 10a(35a/(12a-55))
The value of a is 11.
b = 35(11)/(12(11) – 55))
b = 5
Putting back the equations,
x + y = 11
x – y = 5
Adding up the equations give us,
2x = 16
x = 8 km/hr
The value of x, the speed of the boat in still water, is 8 km/hr.
speed of the stream = 3 km/hr
and speed of boat in still water= 8 km/hr
Step-by-step explanation:Let s be the speed of the boat upstream
and s' be the speed of the boat downstream.
We know that:
[tex]Time=\dfrac{distance}{speed}[/tex]
Hence, we get:
[tex]\dfrac{35}{s}+\dfrac{55}{s'}=12[/tex]
and
[tex]\dfrac{30}{s}+\dfrac{44}{s'}=10[/tex]
Now, let
[tex]\dfrac{1}{s}=a\ and\ \dfrac{1}{s'}=b[/tex]
Hence, we have:
[tex]35a+55b=12--------------(1)\\\\\\and\\\\\\30a+44b=10--------------(2)[/tex]
on multiplying equation (1) by 4 and equation (2) by 5 and subtract equation (1) from (2) we get:
[tex]a=\dfrac{1}{5}[/tex]
and by putting value of a in (2) we get:
[tex]b=\dfrac{1}{11}[/tex]
Hence, speed of boat in upstream= 5 km/hr
and speed of boat in downstream= 11 km/hr
and we know that:
speed of boat in upstream=speed of boat in still water(x)-speed of stream(y)
and speed of boat in downstream=speed of boat in still water(x)+speed of stream(y)
Hence, we get:
[tex]x-y=5\\\\\\and\\\\\\x+y=11[/tex]
Hence, on solving the equation we get:
[tex]x=8[/tex]
and y=3
Hence, we get:
speed of the stream = 3 km/hr
and speed of boat in still water= 8 km/hr
if the divisor is 40 what is the least 3 digit number dividend that would give a remainder of 4
A line passes through (−2,7) and (3,2). Find the slope-intercept form of the equation of the line. Then fill in the value of the slope, m, and the value of the y-intercept
The equation of line passing through points (-2,7) and (3,2) is y = -x + 5. The slope, m, is -1. The y-intercept, b, is 5.
Explanation:The subject matter here is finding the equation of a line in slope-intercept form, which is expressed as y = mx + b. Here, 'm' is the slope of the line and 'b' is the y-intercept. The slope, m, can be found using the formula: m = (y2 - y1)/(x2 - x1). Applying the coordinates given, (-2,7) and (3,2), we find the slope, m = (2 - 7) / (3 - (-2)) = -5 / 5 = -1.
Then, substituting m, x, and y into the equation, we get the y-intercept. Using the point (-2,7), we have: 7 = -1*-2 + b -> 7 = 2 + b -> b = 7 - 2 = 5. So the y-intercept, b, is 5. Therefore, the equation of the line in slope-intercept form is y = -x + 5.
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1 1 2 4 3 9 4 what is the next number
If log63+log672=x, what is the value of x?
If log65 = 1.812, what is the value of log 1000 65? a. 0.1812 b. 0.00182 c. 0.604 d. 0.0604
Answer:
0.604 on a p e x
A group of 11 friends ordered four pizzas to share. They divided the pizzas up evenly and all ate the same amount. Express in decimal form the proportion of a pizza that each friend ate.
Final answer:
Each friend ate approximately 0.3636 of a pizza when the four pizzas were divided evenly among 11 friends.
Explanation:
The student's question involves dividing four pizzas evenly among 11 friends, so each person gets the same proportion of pizza. We need to convert this proportion into decimal form to answer the question.
To find the proportion of a pizza that each friend ate, we calculate 4 pizzas ÷ 11 friends. So, each friend ate ≈ 0.3636 of a pizza. We arrived at this by dividing 4 by 11, which yields a repeating decimal, so we round it to four decimal places to express it accurately.
This value represents the proportion of pizza each person ate when the pizzas were divided equally.
jim is running on a trail that is 5/4 of a mile long. so far he has run 2/3 of the trail. how many miles has he run so far
What the answer to this question?
volume = (1/3)*PI*(r1^2+r1*r2+r2^2)*h
h=10
r1=5
r2=2
= 408.41 cubic inches
round off answer as needed.
Adam can spend a maximum of $252 on office supplies. Each ream of paper costs $6. Each ink cartridge costs $18. Which of the following graphs represents the possible combinations of paper and ink cartridges that he may buy? *Graph pictures below*
Answer:
Option A The graph in the attached figure
Step-by-step explanation:
Let
x-----> the number of ream of paper
y-----> the number of ink cartridge
we know that
[tex]6x+18y\leq 252[/tex] ----> inequality that represent the possible combinations of paper and ink cartridges that Adam may buy
using a graphing tool
the solution is the triangular shaded area
see the attached figure