For a project, you cut paper into strips that are 2 1/2 inches long by 3/4 inches wide. what is the area of each strip of paper
The area of each strip of paper is 15/8 square inches.
Explanation:The area of each strip of paper can be calculated by multiplying the length and width of the strip. In this case, the length is 2 1/2 inches and the width is 3/4 inches. To multiply fractions, multiply the numerators together to get the numerator of the product, and multiply the denominators together to get the denominator of the product. So, the area of each strip is:
(2 1/2 inches) x (3/4 inches) = (5/2 inches) x (3/4 inches) = 15/8 square inches.
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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4,032 divided by 79 = ------
A positive integer is 5 more than 23 times another. Their product is 6732. Find the two integers.
A real estate agent sells a house for $92,000. She receives a 6% commission on the sale of the house.
How much did she earn on the sale?
Round your answer to the nearest cent.
need answer in 10 mins. or less
Amount earn as commission on real state building is $5,520
Commission based problem:What information do we have?
Amount of real state house = $92,000
Percentage of Commission on sale = 6%
Amount earn as commission = Amount of real state house × Percentage of Commission on sale
Amount earn as commission = 92,000 × 6%
Amount earn as commission = 92,000 × 0.06
Amount earn as commission = $5,520
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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?
Of all the species in the world, 444 out of every 555 are insects. What percentage of species are insects?
37.5% of what number is 57?
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.
A city council consists of six Democrats and six Republicans. If a committee of seven people is selected, find the probability of selecting four Democrats and three Republicans.
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.Find the inverse function of f informally.
.A new truck costs $32,000. The car’s value will depreciate over time, which means it will lose value. For tax purposes, depreciation is usually calculated linearly. If the truck is worth $24,500 after three years, write an explicit formula for the value of the car after n years.
The explicit formula for the value of the truck after [tex]\(n\)[/tex] years is:
[tex]\[V(n) = -2,500n + 32,000\][/tex]
Where [tex]\(V(n)\)[/tex] represents the value of the truck after [tex]\(n\)[/tex] years.
To write an explicit formula for the value of the truck after n years with linear depreciation, we can use the formula for the equation of a straight line:
[tex]\[y = mx + b\][/tex]
Where:
- [tex]\(y\)[/tex] is the value of the truck after [tex]\(n\)[/tex] years.
- [tex]\(m\)[/tex] is the slope of the line, which represents the rate of depreciation per year.
- [tex]\(x\)[/tex] is the number of years.
-[tex]\(b\)[/tex] is the initial value of the truck.
Given:
- Initial value [tex](\(b\)): $32,000[/tex]
- Value after 3 years [tex](\(y\)): $24,500[/tex]
We can use these values to find the slope [tex](\(m\))[/tex]:
[tex]\[m = \frac{y_2 - y_1}{x_2 - x_1}\][/tex]
Substituting the given values:
[tex]\[m = \frac{24,500 - 32,000}{3 - 0}\]\[m = \frac{-7,500}{3}\]\[m = -2,500\][/tex]
Now, we have [tex]\(m = -2,500\)[/tex]. We can substitute this slope and the initial value into the equation:
[tex]\[y = -2,500x + 32,000\][/tex]
The explicit formula for the value of the truck after [tex]\(n\)[/tex] years is:
[tex]\[V(n) = -2,500n + 32,000\][/tex]
Where [tex]\(V(n)\)[/tex] represents the value of the truck after [tex]\(n\)[/tex] years.
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 restaurant offers
7
appetizers and
9
main courses. In how many ways can a person order a two-course meal?
Using the concept of combinations, the person can order a two course meal in 63 ways
How many ways can a person order a two course meal?To determine the number of ways a person can order a two-course meal from the given options of appetizers and main courses, we can use the concept of combinations.
In this case, we need to choose one appetizer and one main course. Since these choices are independent, we can use the principle of multiplication.
The number of ways to choose one appetizer from 7 options is 7, and the number of ways to choose one main course from 9 options is 9.
Therefore, the total number of ways to order a two-course meal is given by the product of these two choices:
Number of ways = 7 * 9 = 63.
Hence, a person can order a two-course meal in 63 different ways from the given options of appetizers and main courses.
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Ellen deposits $6,773 into an account earning 1% annually. After seven years what will Ellen's balance have grown to, including interest?
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?
Find the point on the line y = 4x + 5 that is closest to the origin.
Step 1
Find the equation of the line perpendicular to [tex] y = 4x + 5 [/tex] that pas through the origin
we know that
the slope of the equation [tex] y = 4x + 5 [/tex] is
[tex] m1=4 [/tex]
if two lines are perpendicular
then
the product of their slopes is equal to [tex] -1 [/tex]
[tex] m1*m2=-1 [/tex]
the slope of the line perpendicular is equal to
[tex] m2=-\frac{1}{m1} \\\\ m2=-\frac{1}{4} [/tex]
with m2 and the origin find the equation of the line
[tex] y-y1=m(x-x1)\\ y-0=(-\frac{1}{4} )*(x-0)\\ y=-\frac{1}{4} x [/tex]
Step 2
Solve the system
[tex] y = 4x + 5 [/tex]---> equation [tex] 1 [/tex]
[tex] y=-\frac{1}{4} x [/tex]-----> equation [tex] 2 [/tex]
Multiply equation [tex] 1 [/tex] by [tex] -1 [/tex]
Adds equation [tex] 1 [/tex] and equation [tex] 2 [/tex]
[tex] -y = -4x - 5 [/tex]
[tex] y=-\frac{1}{4} x \\ ----- [/tex]
[tex] 0=-4x-\frac{1}{4} x-5\\ \\ \frac{17}{4} x=-5\\ \\ x=-\frac{20}{17} [/tex]
[tex] y = 4x + 5 [/tex]
[tex] y = 4*(-\frac{20}{17}) + 5 \\ \\ y=\frac{5}{17} [/tex]
the solution is the point [tex] (-\frac{20}{17} ,\frac{5}{17} ) [/tex]
[tex] (-\frac{20}{17} ,\frac{5}{17} ) [/tex]=[tex] (-1.176 ,0.294 ) [/tex]
therefore
the answer is
the point is [tex] (-\frac{20}{17} ,\frac{5}{17} ) [/tex]
see the attached figure
The point [tex]\boxed{(-1.176,0.294)}[/tex] on the line [tex]y=4x+5[/tex] is the closest point to the origin.
Further explanation:
The general form of linear function is as follows:
[tex]\boxed{y=mx+c}[/tex]
A linear function has one independent variable and one dependent variable. The independent variable is [tex]x[/tex] and the dependent variable is [tex]y[/tex].
Here, [tex]c[/tex] is the constant term and [tex]m[/tex] is the slope and gives the rate of change of dependent variable.
The point slope form of a line is given as follows:
[tex]\boxed{y-y_{1}=m(x-x_{1})}[/tex]
where [tex](x_{1},y_{1})[/tex] is the point on the line and [tex]m[/tex] is the slope of the line.
It is given that the equation of the line is as follows:
[tex]y=4x+5[/tex]
where [tex]4[/tex] is the slope of the line.
Consider the slope of the line [tex]y=4x+5[/tex] as [tex]m_{1}=4[/tex].
We first find the line perpendicular to the line [tex]y=4x+5[/tex] that passes through the origin as shown in Figure 1 in the attachment below.
If two lines are perpendicular then the product of their slopes is [tex]-1[/tex] that is [tex]m_{1}\tiimes m_{2}=-1[/tex].
And [tex]m_{2}[/tex] is the slope of the perpendicular line.
Calculated the value of [tex]m_{2}[/tex] as follows:
[tex]\begin{aligned}m_{1}\times m_{2}&=-1\\m_{2}&=-\dfrac{-1}{m_{1}}\\&=\dfrac{-1}{4}\\&=-0.25\end{aligned}[/tex]
Therefore, the slope of perpendicular line is [tex]-0.25[/tex].
We have the point [tex](0,0)[/tex] on the line and the slope of the line.
Thus, the equation of line is,
[tex]\begin{aligned}y-y_{1}&=m_{2}(x-x_{1})\\y-0&=(-0.25)(x-0)\\y&=-0.25x\end{aligned}[/tex]
The equation of the perpendicular line is as follows:
[tex]\boxed{y=-0.25x}[/tex] .......(2)
Substitute [tex]y=-0.25x[/tex] in equation (1).
[tex]\begin{aligned}-\dfrac{x}{4}&=4x+5\\-\dfrac{x}{4}-4x&=5\\-\dfrac{17x}{4}&=5\\-17x&=5\times 4\\x&=-\dfrac{20}{17}\\x&\approx-1.176\end{aligned}[/tex]
Now, put the value of [tex]x[/tex] in equation (2) to get the value of [tex]y[/tex] as,
[tex]\begin{aligned}y&=-0.25\times (-1.176)\\&=0.294\end{aligned}[/tex]
Therefore, the closest point is [tex]\boxed{(-1.176,0.294)}[/tex].
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Answer details:
Grade: High school
Subject: Mathematics
Chapter: Linear equations
Keywords: Linear equations, slope of a line, equation of the line, function, real numbers, ordinates, abscissa, interval, open interval, closed intervals, semi-closed intervals, semi-open intervals, sets, range domain, codomain.
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?
The length of a rectangle is 10 meters more than the width. If the perimeter is 184 meters, what are the length and the width?
e figure below shows the derivative F' of a function F. If F(20) equals 100, estimate the maximum value attained by by F
Where the above figure is given as the attendant derivatives F', the maximum value attained by F is 400.
How to obtain the maximum value
It is to be noted that:
Max occurs at x=50 where F'(x) = 0.
We need to find value for F(50) using fundamental theorem of calculus:
[tex]\int_{50}^{50} F' dx = F(50) - F(20)[/tex]
= F (50) -100
F(50) = 100 + A
Where A is the estimated Area under the graph.
If we assume the shaded region is close to a right triangle then the Area is
(1/2) x 30 x 20
= 300
Thus,
F (50) = 100 + 300
= 400
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Ex 9-9 Straight-line depreciation ObJ. 2
A refrigerator used by a meat processor has a cost of $48,000, an estimated residual value of $9,000, and an estimated useful life of 15 years. What is the amount of the annual depreciation computed by the straight-line method?
The annual depreciation expense for the meat processor's refrigerator, using the straight-line method, is $2,600.
Straight-line Depreciation Calculation
The straight-line depreciation method spreads the cost of an asset evenly over its useful life. To calculate the annual depreciation expense for the refrigerator, we follow this formula: (Cost of the asset - Residual value) / Estimated useful life.
In this case, the numbers provided are a cost of $48,000, a residual value of $9,000, and an estimated useful life of 15 years. Using these values, the calculation is as follows:
(48,000 - 9,000) / 15 = 39,000 / 15 = $2,600 per year.
Therefore, the annual depreciation expense that the meat processor would record for this refrigerator is $2,600.
An assembly-line machine turns out washers with the following thicknesses (in millimeters). Find the mean and standard deviation of the thicknesses.
1.42, 1.83, 1.43, 1.87, 1.22, 1.44, 1.72, 1.91, 1.42, 1.87, 1.42, 1.99, 1.27, 1.94, 1.67, 1.22
The mean thickness of the washers is 1.55 mm and the standard deviation is approximately 0.296 mm.
Explanation:To find the mean of the thicknesses, we add up all the values and divide it by the total number of values. In this case, the sum of the thicknesses is 24.81. Dividing this by the total number of values, which is 16, gives us a mean of 1.55 mm.
To find the standard deviation, we need to calculate the variance first. The formula for variance is sum of (value - mean)^2 divided by n - 1, where n is the number of values. After calculating the variance, we take its square root to find the standard deviation. In this case, the variance is 0.0877, and the standard deviation is approximately 0.296 mm.
I need help finding the reference angles for these angles.
1) 1.4
2) [tex]-1.4 \pi [/tex]
3) 2.3
4) [tex]2.3 \pi [/tex]
There are seven ice skaters in a competition. How many different possibilities are there for the top three places? (A different ordering of the same three skaters counts as a different possibility.)
What is the probability that a couple will have a girl a boy a girl and a boy in this specific order?
The probability that a couple will have a girl a boy a girl and a boy in this specific order is 1/16
Probability is the likelihood or chance that an event will occur.
Probability = Expected outcome/Total outcome
If the couple will give birth to a boy and a girl, hence the total outcome of the event is 2
Pr(having a boy) = 1/2
Pr(having a girl) = 1/2
Pr(a girl, a boy, a girl and a boy) = 1/2 * 1/2 *1/2 * 1/2
Pr(a girl, a boy, a girl, and a boy) = 1/16
Hence the probability that a couple will have a girl a boy a girl and a boy in this specific order is 1/16
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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
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.
in a school election 3/4 of the students vote. There are 1464 ballots. write and solve an equation to find the number of students.
PLEASE HELP ;(
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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