Suzie hangs 10 pictures that each measure 8.5 inches wide on a wall. She spaces them 6 inches apart with 2 inches on each end, so that they fit perfectly on the wall. How wide is the wall?
A certain radioactive isotope has leaked into a small stream. Four
hundred days after the leak, 13% of the original amount of the substance remained. Determine the half-life of this radioactive isotope.
The half-life of this radioactive isotope is 135.897 days and this can be determined by using the formula of half-life.
Given :
A certain radioactive isotope has leaked into a small stream.Four hundred days after the leak, 13% of the original amount of the substance remained.The formula of the half-life is given by:
[tex]\rm A =P \left(\dfrac{1}{2} \right)^{\dfrac{t}{t_{1/2}}}[/tex]
where A is the final concentration, P is the initial concentration, t is the time period, and [tex]\rm t_{1/2}[/tex] is the half-life.
Now, substitute the known values in the above formula.
[tex]\rm 0.13P =P \left(\dfrac{1}{2} \right)^{\dfrac{400}{t_{1/2}}}[/tex]
Simplify the above expression.
[tex]\rm 0.13 = \left(\dfrac{1}{2} \right)^{\dfrac{400}{t_{1/2}}}[/tex]
Take the log on both sides.
[tex]\rm log(0.13) = {\dfrac{400}{t_{1/2}}} \times log\left(\dfrac{1}{2} \right)[/tex]
[tex]\rm t_{1/2} = \dfrac{400\times log(0.5)}{log(0.13)}[/tex]
Simplify the above expression in order to evaluate the [tex]\rm t_{1/2}[/tex].
[tex]\rm t_{1/2} = 135.897\;days[/tex]
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The half-life of a radioactive isotope is calculated by using the decay formula and solving for the known half-life, given that 13% of the original isotope remained after 400 days.
Explanation:To determine the half-life of a radioactive isotope, we first need to understand that half-life is the time required for half of a sample of a radioactive isotope to decay. From the problem, it stated that after 400 days, only 13% of the original isotope remained. The formula for decay is N = N0 * (1/2)^(t/T), where N is the final amount, N0 is the initial amount, t is the time elapsed, and T is the half-life.
Given that N/N0 is 0.13 after 400 days, we can rewrite the decay formula to solve for T. By substituting the values into the equation, i.e., 0.13 = (1/2)^(400/T), we can use the laws of logarithms to solve for T. So then we will get the half-life (T) of the isotope.
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The Extreme Rock Climbing Club planned a climbing expedition. The total cost was $1400, which was to be divided equally among the members going. Prior to the trip, 3 members decided not to go. If the cost per person increased by $105, how many people went on the expedition?
Answer:
5 people
Step-by-step explanation:
Let
x = number of peopley = cost per personThe total cost was $1400, which was to be divided equally among the members going. Then,
y = 1400 / x [1]
Prior to the trip, 3 members decided not to go. The cost per person increased by $105.
y + 105 = 1400 / (x - 3)
(y + 105) . (x - 3) = 1400 [2]
We replace [1] in [2],
(1400/x + 105) . (x - 3) = 1400
1400 - 4200/x + 105x -315 = 1400
- 4200/x + 105x = 315
(- 4200 + 105x²)/x = 315
- 4200 + 105x² = 315 x
105x² - 315x - 4200 = 0
When we solve this quadratic equation using the quadratic formula, we get x₁ = 8 and x₂ = -5. Only the positive x has sense.
The people that went to the expedition was x - 3 = 8 - 3 = 5.
Find a trivial lower-bound class for each of the following problems and indicate, if you can, whether this bound is tight.
a. finding the largest element in an array
b. checking completeness of a graph represented by its adjacency matrix
c. generating all the subsets of an n-element set
d. determining whether n given real numbers are all distinct
The trivial lower-bound classes for the problems provided include O(n) for finding the largest element in an array, O(n^2) for graph completeness, O(2^n) for subset generation, and O(n log n) for distinctness checking. Some of these bounds are tight, such as O(n) and O(2^n), while others may not be.
Explanation:For each of the following problems, we can establish a trivial lower-bound class and discuss whether it is tight:
Finding the largest element in an array: The trivial lower-bound class is O(n), as we must inspect each element at least once. This bound is tight because it's not possible to determine the largest element without looking at all elements.Checking completeness of a graph represented by its adjacency matrix: The trivial lower-bound class is O(n2), since we must examine all n2 possible edges for an n-vertex graph. However, this is not tight because we could determine non-completeness before examining all edges if a missing edge is found.Generating all the subsets of an n-element set: The lower-bound class is O(2n), because there are 2n possible subsets. This is a tight bound since we cannot generate less than all possible subsets.Determining whether n given real numbers are all distinct: The trivial lower-bound class is O(n log n), assuming we sort and then do a linear scan for duplicates. It might not be tight since there may be a method to do it in linear time, but it's unknown.37.5% of what number is 57?
One person in a stadium filled with 100,000 people is chosen at random to win a free pair of airline tickets. what is the probability that it will not be you?
Find MF if M is the midpoint of CF for the points C(3,7)F(5,5)
Solve the equation by factoring. (Enter your answers as a comma-separated list. Let k be any integer. Round terms to three decimal places where appropriate. If there is no solution, enter NO SOLUTION.) 4 sin(θ) cos(θ) + 2 sin(θ) − 2 cos(θ) − 1 = 0
Jeremy can type 140 words in 4 minutes and 525 words in 15 minutes. Let x= minutes spent typing and y= words typed. what is the rate of change of y with respect to x
The rate of change of y with respect to x is 35 words per minute.
Explanation:The rate of change of y with respect to x can be found by taking the slope of the line that represents the relationship between x and y. To find the slope, we can use the formula:
slope = change in y / change in x
Using the given information, we can calculate the slope as:
slope = (525 - 140) / (15 - 4) = 385 / 11 = 35
Therefore, the rate of change of y with respect to x is 35 words per minute.
Final answer:
Jeremy's rate of change in typing speed is 35 words per minute, calculated by dividing the change in words typed by the change in time (385 words/11 minutes).
Explanation:
The student is asking about the rate of change of the words typed with respect to time spent typing. To calculate this, we will use the given values: 140 words in 4 minutes and 525 words in 15 minutes. The rate of change, essentially being the slope of the line that represents the relationship between words typed (y) and the time (x), can be calculated by finding the ratio of the change in words to the change in time:
Firstly, calculate the change in words typed (Δy) which is 525 - 140 = 385 words.Then, calculate the change in time (Δx) which is 15 - 4 = 11 minutes.The rate of change (slope) is Δy/Δx which equals 385 words / 11 minutes = 35 words per minute.This means that Jeremy types at an average speed of 35 words per minute.
This is a mathematical sentence written with a greater than, a less than sign, or a NOT equal to sign.
The mathematical sentence written with a greater than, a less than sign, or a not equal sign is know as Inequality.
What is Inequality?Inequalities are mathematical formulas in which neither side is equal. Unlike equations, we compare two values in inequality. In between, the equal sign is substituted by a less than (or less than or equal to), greater than (or greater than or equal to), or not equal to sign.
As, the mathematical sentence written with a greater than, a less than sign, or a not equal sign.
The definition of inequality is that two things are NOT equal. One of the items could be less than, greater than, less than or equal to the other things, or greater than or equal to the other things.
So, the statement is Inequality statement.
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find the least LCM of 28 and 98 to the 2nd power
Melissa's coffee shop makes a blend that is a mixture of two types of coffee type a coffee cost Melissa for 4.75 per pound and type B coffee costs 5.80 per pound this month the Melissa made 123 pounds of the blend for a total cost of $643.05 how many pounds of type B coffee did she use?
At a concession stand, three
hot dogs
and two
hamburgers
cost $7.75
;
two
hot dogs
and three
hamburgers
cost $8.50
.
Find the cost of one hot dog
and the cost of one hamburger
.
To find the individual cost of a hot dog and hamburger from a concession stand, we set up a system of equations from the given information and solve it to get the cost of one hot dog as $1.25 and one hamburger as $2.00.
To find the cost of one hot dog and one hamburger, we set up a system of equations based on the information given:
3 hot dogs + 2 hamburgers = $7.752 hot dogs + 3 hamburgers = $8.50Let's denote the cost of one hot dog as d and the cost of one hamburger as h. We can then rewrite the given information as:
3d + 2h = 7.752d + 3h = 8.50Solving this system of equations, we can multiply the first equation by 2 and the second equation by 3, and subtract them, cancelling out the hamburger terms:
6d + 4h = 15.506d + 9h = 25.50Subtracting the first equation from the second gives:
0d + 5h = 10.00Thus, one hamburger costs $2.00. Plugging this back into the first equation, we get:
3d + 2(2) = 7.75Which simplifies to:
3d + 4 = 7.753d = 3.75d = 1.25Therefore, one hot dog costs $1.25.
A distance of 6.5 x 10^-8 is multiplied by 10. The result is written in scientific notation. What is the new exponent
The new exponent of the product is -7
How to determine the new exponentFrom the question, we have the following parameters that can be used in our computation:
6.5 x 10^-8 is multiplied by 10.
This means that
Product = 6.5 x 10^-8 * 10
Evaluate the product
So, we have
Product = 6.5 x 10^-7
From the above, we have
Exponent = -7
Hence, the new exponent is -7
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A comic book costs $3.99, including tax. It is marked up 45%.
What is the final price of the comic book after it is marked up?
You are a farmer and want to spend under $35,000 on farm equipment. You need a hay bailer that costs $6,250 and several plowing disks cost $2,500 each. Write an inequality that models how many plowing disks could be purchased within your budget. What is the maximum number of plowing disks you can buy?
Answer in factored form 3x^2+21x+36
Find the point on the hyperbola xy=8 that is closet to the point (3,0)
The point that is closest to the point (3, 0) can be found by finding the
minimum value for the function of the distance between points.
The closest point on the hyperbola x·y = 8 to the point (3, 0) is the point [tex]\underline{(4, \ 2)}[/tex]
Reasons:
The equation of the hyperbola is x·y = 8
The given point = (3, 0)
Therefore;
[tex]y = \dfrac{8}{x}[/tex]
The distance of a point from a line, [tex]d = \mathbf{ \sqrt{\left (y_{2}-y_{1} \right )^{2}+\left (x_{2}-x_{1} \right )^{2}}}[/tex]
Where;
(x₁, y₁) = (3, 0)
(x₂, y₂) = The closest point on the hyperbola
We get;
[tex]d = \sqrt{\left (y-0 \right )^{2}+\left (x-3 \right )^{2}}[/tex]
Which gives;
[tex]d = \sqrt{\left (\dfrac{8}{x} -0 \right )^{2}+\left (x-3 \right )^{2}} = \mathbf{ \sqrt{\left (\dfrac{8}{x}\right )^{2}+\left (x-3 \right )^{2}}}[/tex]
At the minimum distance, using an online app, we have;
[tex]\dfrac{d}{dx}d = \dfrac{d}{dx} \left( \sqrt{\left (\dfrac{8}{x}\right )^{2}+\left (x-3 \right )^{2}} \right) = \dfrac{-64 + x^3 \cdot (x - 3)}{x^2 \cdot \sqrt{x^2 \cdot (x - 3)^2 + 64} } = 0[/tex]
Which gives;
x³·(x - 3) - 64 = 0
Using a graphing calculator, we get;
(x - 4)·(x³ + x² + 4·x + 16) = 0
Therefore;
A solution for the closest or point of minimum distance is x = 4
Which gives;
[tex]y = \dfrac{8}{4} = 2[/tex]
Therefore, the closest point on the hyperbola x·y = 8 to the point (3, 0) is the
point [tex]\underline{(4, \ 2)}[/tex]
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quadratic formula for 2x^2-x-3=0
lettuce is packaged in four bags to a box. if there are 3.5 boxes of lettuce in a cooler, how many bags of lettuce would there be?
Answer:
There are 14 bags in 3.5 boxes.
Step-by-step explanation:
Given : Lettuce is packaged in four bags to a box. If there are 3.5 boxes of lettuce in a cooler.
To find : How many bags of lettuce would there be?
Solution :
Lettuce is packaged in four bags to a box.
i.e, In 1 box there are 4 bags.
If there are 3.5 boxes of lettuce in a cooler.
In 3.5 boxes number of bags are [tex]4\times 3.5=14[/tex]
Therefore, There are 14 bags in 3.5 boxes.
Da sofa is on sale for $98.60, which is 29% of the regular price. What is the regular price ?
Of all the species in the world, 444 out of every 555 are insects. What percentage of species are insects?
One integer is 5 more than another. Their product is 126. Find the integers.
The perimeter of a rectangle is 36 meters. The length is 2 meters more than three times the width find the length of the rectangle
The length of the rectangle with a perimeter of 36 meters and a length that is 2 meters more than three times the width is 14 meters.
The perimeter of a rectangle is twice the sum of its length and width. According to the question, the perimeter of the rectangle is 36 meters, and the length (L) is 2 meters more than three times the width (W). We can express this relationship as follows: L = 3W + 2.
The formula for the perimeter (P) of a rectangle is P = 2L + 2W.
Using the given perimeter of 36 meters, we can set up the equation as 2(3W + 2) + 2W = 36.
Simplifying this equation, we find that 8W + 4 = 36, and thus W = 4 meters.
Substituting the value of W back into the equation for L gives us L = 3(4) + 2, which means the length of the rectangle is 14 meters.
Find the inverse function of f informally.
A person can purchase a particular model of a new car with a choice of ten colors, with or without automatic transmission, with or without four-wheel drive, with or without air conditioning, and with two, three, or four radio-CD speakers. How many different options are there for this model of car?
To find the different options for a car model with multiple customizable features, you multiply the number of choices for each feature. For this model, with 10 colors, 2 options each for transmission, four-wheel drive, and air conditioning, and 3 speaker options, there are 240 different combinations.
Explanation:To calculate the number of different options for this model of a car, we need to consider all the features that can be selected and multiply the number of choices for each feature together. For the color, there are 10 options. For each of the features, such as automatic transmission, four-wheel drive, and air conditioning, there are 2 choices: with or without. Finally, for the radio-CD speakers, there are 3 options: two, three, or four speakers. Multiplying these choices together gives us the total number of different options:
10 (colors) × 2 (transmission) × 2 (four-wheel drive) × 2 (air conditioning) × 3 (radio-CD speakers) = 240 different options.
So, there are 240 different options available for this model of a car.
Final answer:
To find the total number of different options for the car, we multiply the number of choices for each feature resulting in 240 different options.
Explanation:
The student is asking about the number of different option combinations for a specific model of a new car. The car can be customized with ten colors, automatic transmission or not, four-wheel drive or not, air conditioning or not, and can have two, three, or four radio-CD speakers. To determine the total number of different options, we multiply the number of choices for each feature:
10 colors2 options for automatic transmission (with or without)2 options for four-wheel drive (with or without)2 options for air conditioning (with or without)3 options for the number of radio-CD speakers (two, three, or four)Therefore, the total number of different options is calculated as follows:
10 (colors) × 2 (transmission) × 2 (drive) × 2 (air conditioning) × 3 (speakers)
= 240 different options.
brent counted 10 red cards, 10 black cards and 20 blue cards. what is the ratio of red cards to other cards
Final answer:
The ratio of red cards to other cards that Brent counted (10 red cards, 10 black cards, 20 blue cards) is 1:3 after summing black and blue cards to find the total number of other cards and simplifying the ratio.
Explanation:
Brent counted 10 red cards, 10 black cards and 20 blue cards. The question is what is the ratio of red cards to other cards. To calculate the ratio, we need to determine the total number of cards that are not red, which would be the black and blue cards combined.
First, we sum the black and blue cards:
10 black cards + 20 blue cards = 30 other cards.
Now, we set up the ratio of red cards to other cards:
10 red cards : 30 other cards, which simplifies to 1 : 3 when both sides of the ratio are divided by 10, the greatest common divisor.
Therefore, the ratio of red cards to other cards is 1:3.
Tom and Jim are distant cousins who meet during their family Thanksgiving meal. Tom tells Jim that he has 3 kids, and the product of their ages is 72. He also tells him the sum of their ages. Jim says he still can’t correctly guess the ages of Tom's kids. Finally, Tom says, "My youngest child's name is Joy." Jim can then correctly guess. What are the ages?
Answer:
6, 6, 2
Step-by-step explanation:
Factor 72:
72 = 2³ × 3²
Look for groups of 3 numbers whose products are 72 and their sums:
2, 4, 9: sum 15
8, 9, 1: sum 18
8, 3, 3: sum 14
6, 6, 2: sum 14
6, 4, 3: sum 13
12, 6, 1: sum 19
12, 3, 2: sum 17
18, 4, 1: sum 23
18, 2, 2: sum 22
24, 3, 1: sum 28
36, 2, 1: sum 39
72, 1, 1: sum 74
Jim knows the sum, but he can't guess the ages. that means the sum must be 14 for which there are two sets of ages which have a product of 72:
8, 3, 3 and 6, 6, 2
Then when he says "my youngest", that means one age is less than the other two. The only choice is 6, 6, 2.
A ladder is leaning against a building so that the distance from the ground to the top of the ladder is 2 feet less than the length of the ladder. Find the length of the ladder if the distance from the bottom of the ladder to the building is 6 feet.
To find the length of the ladder, we can use the property of right triangles. The length of the ladder is 10 feet.
Explanation:To find the length of the ladder, we can use the property of right triangles. Let's assume the length of the ladder is 'L'.
According to the given information, the distance from the ground to the top of the ladder is 2 feet less than the length of the ladder. So, the height of the triangle formed by the ladder, the building, and the ground is L - 2.
The distance from the bottom of the ladder to the building is given as 6 feet. Using the Pythagorean theorem, we can set up the following equation: (L-2)^2 + 6^2 = L^2
Simplifying the equation, we get L^2 - 4L + 4 + 36 = L^2, which simplifies further to 40 = 4L. Dividing both sides by 4, we get L = 10.
Therefore, the length of the ladder is 10 feet.
Between which two ordered pairs does the graph of f(x) = x2 + x – 9 cross the negative x-axis?
Quadratic formula: x =
(–6, 0) and (–5, 0)
(–4, 0) and (–3, 0)
(–3, 0) and (–2, 0)
(–2, 0) and (–1, 0)
we have that
[tex] f(x) =x^{2} + x - 9 [/tex]
Using a graph tool
see the attached figure
The function represent a vertical parabola that open up
the vertex is a minimum-------> is the point [tex] (-0.5,-9.3) [/tex]
The x-intercepts are the points when the y coordinate is equal to zero
The x-intercepts are the points [tex] (-3.5,0) [/tex] and [tex] (2.5,0) [/tex]
so
the function cross the negative x-axis at point [tex] (-3.5,0) [/tex]
therefore
the answer is
(–4, 0) and (–3, 0)