Answer:
break even units for both the cases will be 5
Step-by-step explanation:
Data provided in the question:
For the case 1
Variable cost = $20 each
Selling cost = $50
Rent for the booth fair = $150
Now,
Let break even units be x
At break even
Total cost = Total revenue
Thus,
$20x + $150 = $50x
or
$50x - $20x = $150
or
$30x = $150
or
x = 5
Case 2
Variable cost = $15 per unit
Thus,
At break even
Total cost = Total revenue
Thus,
$15x + $150 = $50x
or
$50x - $15x = $150
or
$35x = $150
or
x = 4.28 ≈ 5
The break even point will still remain the same.
The break-even point is calculated by setting total cost equal to total revenue and solving for the number of units produced and sold (denoted as 'units').
Given the current variable cost per unit ($20), the sale price per unit ($50), and the fixed cost (booth rent - $150), we can set up the equation as follows:
Total Cost = Fixed cost (booth rent) + variable cost per unit * units
Total Revenue = sale price per unit * units
Setting these two equal to each other, we get:
150 + 20*units = 50*units
By rearranging this equation, we find:
units = 150 / (50 - 20)
This calculates out to 5 units. Therefore, Ray needs to sell 5 units to break even with his current costs.
If Ray is able to reduce his variable cost to $15 per unit, we will repeat the same calculation with the new variable cost:
units = 150 / (50 - 15)
This calculates out to approximately 4.29 units. Since Ray cannot sell a fraction of a unit, he would have to sell 5 units to fully cover his costs, but he would begin to make a profit sooner than with his current variable cost. In fact, from the 5th unit sold, part of the revenue would go towards profit. Therefore, with the reduced variable cost, his break-even point would be closer to 4 units, but practically still 5 units.
In conclusion, with his current costs, Ray's break-even point is at 5 units. If he is able to reduce his variable cost to $15 per unit, his break-even point would theoretically be lower at approximately 4.29 units, but practically still would round up to 5 units.
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Joel has a goal to practice his clarinet for 4 1/2 per week. The list below shows the number of hours Joel has a practiced so far for the week. Monday 1 1/2 hours Wednesday 1 1/4 hours Thursday 1 hour How many more hours does he need to practice this week to meet his goal
0.75 hours more is needed to practice to meet his goal
Solution:
Given that,
Goal per week of Joel = [tex]4\frac{1}{2} = \frac{2 \times 4 + 1}{2} = \frac{9}{2}[/tex]
From given,
The list below shows the number of hours Joel has a practiced so far for the week
[tex]Monday = 1\frac{1}{2}\ hours = \frac{3}{2}\ hours[/tex]
[tex]Wednesday = 1\frac{1}{4}\ hours = \frac{5}{4}\ hours\\\\Thursday = 1\ hour[/tex]
How many more hours does he need to practice this week to meet his goal
Find the difference
Hours needed = Goal per week of Joel - (monday + wednesday + thursday)
[tex]Hours\ needed = \frac{9}{2} - (\frac{3}{2} + \frac{5}{4} + 1)\\\\Hours\ needed = 4.5-(1.5+1.25+1)\\\\Hours\ needed = 4.5 - 3.75 = 0.75\\\\Hours\ needed = 0.75 = \frac{3}{4}[/tex]
Thus 0.75 hours more is needed to practice to meet his goal
Joel needs to practice \( \frac{3}{4} \) hours more this week to meet his goal.
First, we need to calculate the total number of hours Joel has practiced so far. We will add the hours practiced on Monday, Wednesday, and Thursday.
[tex]Monday: \( 1 \frac{1}{2} \) hours = \( 1 + \frac{1}{2} \) hours = \( \frac{3}{2} \) hours[/tex]
[tex]Wednesday: \( 1 \frac{1}{4} \) hours = \( 1 + \frac{1}{4} \) hours = \( \frac{5}{4} \) hours[/tex]
[tex]Thursday: \( 1 \) hour = \( \frac{4}{4} \) hours (to keep the denominator consistent)[/tex]
Now, let's add these hours together:
[tex]\( \frac{3}{2} + \frac{5}{4} + \frac{4}{4} = \frac{6}{4} + \frac{5}{4} + \frac{4}{4} = \frac{15}{4} \) hours[/tex]
Joel's goal is to practice [tex]\( 4 \frac{1}{2} \)[/tex] hours per week, which is [tex]\( 4 + \frac{1}{2} \) hours = \( \frac{8}{2} + \frac{1}{2} \) hours = \( \frac{9}{2} \) hours.[/tex]
To find out how many more hours Joel needs to practice, we subtract the total hours he has already practiced from his goal:
[tex]\( \frac{9}{2} - \frac{15}{4} \)[/tex]
To subtract these fractions, we need a common denominator, which is 4:
[tex]\( \frac{18}{4} - \frac{15}{4} = \frac{3}{4} \) hours[/tex]
Therefore, Joel needs to practice [tex]\( \frac{3}{4} \)[/tex] hours more to meet his weekly goal.
In Don's congruence flowchart for problem 6-29, one of the ovals "AB/FD= 1". In Phil's flowchart, one of the ovals said, "AB=FD". Discuss with with your team whether these ovals say the same thing. Can equality statements like Phil's always be used in congruence flowcharts?
Yes, they are saying the same thing. In fact, if a ratio equals one, it means that numerator and denominator are equal.
This is the reason why you can always use A=B or A/B, as they are totally equivalent.
Rs 144 is distributed among 2 boys and 3 girls in such a way that each girl gets three times as much as each boy gets. Find how much each boy should get.
Answer:
Each boy gets approximately Rs 13.09
Step-by-step explanation:
We are given the following in the question:
Total money = Rs 144
Let x be the amount each girl gets and y be the amount each boy gets.
The money is distributed among 2 boys and 3 girls. Thus, we can write:
[tex]3x + 2y = 144[/tex]
Also, each girl gets three times as much as each boy gets.
Thus, we can write:
[tex]x = 3y[/tex]
Substituting these values, we get,
[tex]3(3y) + 2y = 144\\11y = 144\\\\y =\dfrac{144}{11} \approx 13.09\\\\x = 3(\dfrac{144}{11}) \approx 39.27[/tex]
Thus, each boy gets approximately Rs 13.09
Answer: each boy should get $39.27
Step-by-step explanation:
Let x represent the amount that each boy should get.
Let y represent the amount that each girl should get.
The total amount of money that would be distributed among the 2 boys and 3 girls is 144 Rs. This is expressed as
2x + 3y = 144 - - - - - - - - - - - -1
The money is shared in such a way that each girl gets three times as much as each boy gets. This is expressed as
y = 3x
Substituting y = 3x into equation 1, it becomes
2x + 3 × 3x = 144
2x + 9x = 144
11x = 144
x = 144/11 = 13.09
y = 3x = 3 × 13.09
y = $39.27
True or false? All occurrences of the letter u in "Discrete Mathematics" are lowercase. Justify your answer
The given statement "All occurrences of the letter u in "Discrete Mathematics" are lowercase" is true.
Here's why:
There are no occurrences of the letter "u" in "Discrete Mathematics" at all.
Therefore, the question of whether they are uppercase or lowercase becomes irrelevant due to the absence of the letter itself.
Because the statement involves a vacuous quantification, meaning it deals with an empty set, it automatically becomes true.
In such cases, it doesn't matter what property is being attributed to the empty set because there are no elements for that property to be true or false for.
Joe uses a ladder to reach a window 10 feet above ground. If the ladder is 3 feet away from the wall, show whether a 12 foot ladder is long enough to reach the window.
Answer:
Step-by-step explanation:
Answer: the ladder would be long enough.
Step-by-step explanation:
The ladder forms a right angle triangle with the wall of the building and the ground. The length of the ladder represents the hypotenuse of the right angle triangle. The height from the window to the base of the building represents the opposite side of the right angle triangle.
The distance from the bottom of the ladder to the base of the building represents the adjacent side of the right angle triangle.
To determine if a 12 foot ladder is long enough to reach the window, we would apply Pythagoras theorem which is expressed as
Hypotenuse² = opposite side² + adjacent side²
Let h represent the height that the ladder would get to. Therefore
12² = h² + 3²
144 = h² + 9
h² = 144 - 9 = 135
h = √135 = 11.62 feet
Since the height of the window is 10 feet above the ground, the ladder would be long enough.
Answer:
c.square root of 28
Step-by-step explanation:
A frustum is made by removing a small cone from a similar large cone.Work out the frustum radius of cone=4.5cm radius of frustum=3cm height of the frustum =3cm height of the cone=9cm
The radius of the cone is found by comparing the ratios of heights and radii in a similar triangle. By applying the given values, the radius of the cone is found to be 9 cm.
Explanation:In mathematics, problems related to finding the dimensions of cones and frustums are common. As the problem mentions, a frustum is created when a smaller, similar cone is removed from a larger cone. To calculate the dimensions of the frustum, we use the properties of similar triangles.
For similar triangles:
Base ratios are equal to height ratios.
Therefore:
Height ratio (height of frustum/height of cone) = Base ratio (radius of frustum/radius of cone)
Applying the given values, it becomes:
3/9 = 3/Radius of cone
Hence, the radius of the cone is 9 cm.
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Final answer:
The volume of a frustum can be calculated using the formula [tex]\(V = \frac{1}{3}\pi h (r_{1}^{2} + r_{2}^{2} + r_{1}r_{2})\)[/tex]. Given the dimensions provided for the frustum with the larger cone having a radius of 4.5 cm, the frustum a radius of 3 cm, and height of 3 cm, the volume of the frustum is approximately 134.25π cubic centimeters.
Explanation:
To work out the volume of a frustum that is formed by removing a small cone from a larger, similar cone, you can make use of the formula for the volume of a frustum:
V = [tex]\(\frac{1}{3}\pi h (r_{1}^{2} + r_{2}^{2} + r_{1}r_{2})\),[/tex]
where:
h is the height of the frustum,
r₁ is the radius of the larger base (radius of the original cone),
r₂ is the radius of the smaller base (radius of frustum).
Given:
Radius of the large cone (r₁) = 4.5 cm,
Radius of the frustum (r₂) = 3 cm,
Height of the frustum (h) = 3 cm.
The height of the original cone (9 cm) is not needed to calculate the volume of the frustum. Plugging the given values into the formula:
[tex]\(V = \frac{1}{3}\pi \times 3 \times (4.5^{2} + 3^{2} + 4.5 \times 3)\)[/tex]
[tex]\(V = \pi (20.25 + 9 + 13.5)\)[/tex]
[tex]\(V = \pi (42.75)\)[/tex]
[tex]\(V = 134.25\pi \text{cm}^{3}\)[/tex]
Therefore, the volume of the frustum is approximately 134.25π cubic centimeters.
Kwame's team will make two triangular pyramids to decorate the entrance to the exhibit. They will be wrapped in the same metallic foil. Each base is an equilateral triangle. If the base has an area of about 10.8 square feet, how much will the team save altogether by covering only the lateral area of the two pyramids? The foil costs $0.24 per square foot.
Answer:
Therefore the team saves $5.18.
Step-by-step explanation:
i) there are two triangular pyramids
ii) the base of each pyramid is an equilateral triangle with an area of 10.8 square feet.
iii) the base of the pyramids are not covered with foil.
iv) since there are two pyramids the total area not covered
= 10.8 [tex]\times[/tex] 2 = 21.6 square feet
v) the cost of foil = $0.24 per square foot.
vi) the team save altogether by covering only the lateral area of the two pyramids
= 21.6 [tex]\times[/tex] 0.24 = $5.18
Therefore the team saves $5.18.
What’s the Value for k?
Answer:
Step-by-step explanation:
The sum of the angles on a straight line is 180 degrees. Therefore,
Angle XYZ + angle MYZ = 180
Angle XYZ + 115 = 180
Angle XYZ = 180 - 115 = 65 degrees
The sum of the angles in a triangle is 180 degrees. It means that
Angle XZY + angle YXZ + angle MYZ = 180
Therefore,
4k + 5 + 6k + 10 + 65 = 180
4k + 6k + 5 + 10 + 65 = 180
10k + 80 = 180
10k = 180 - 80 = 100
Dividing the left hand side and the right hand side of the equation by 10, it becomes
10k/10 = 100/10
k = 10
Statistics encompasses all scientific disciplines in which random occurrences are analyzed. In addition, statistics references any random occurrence which is reported using percentages or proportions. True or false?
Answer: FALSE
Step-by-step explanation: Statistics is the Science of collecting, organising,sumarising,analysing information to reach at a reasonable conclusion or outcome. Statistics also helps to measure levels of confidence in any conclusion or outcome.
Statistics has been applied in several fields like Medicine, pharmacy, Engineering,Biology etc it helps in determining especially in numerical representation the impact of certain conditions or activities or information.
Start with x2 + 4x = 12 and complete the square, what is the equivalent equation? A) (x + 2)2 = 16 B) (x + 2)2 = 14 C) (x + 4)2 = 16 D) (x + 4)2 = 28
Answer: The answer is A
Step-by-step explanation: Add the coefficient of x to the both sides.
X²+4x=12
X²+4x+4=12+4
X²+4x+4=16
X²+2x+2x+4=16
X(x+2)+2(x+2)=16
(X+2)(x+2)=16
(X+2)²=16
The equivalent equation is (x+2)²=16
What are quadratic equations?A quadratic equation can be written in the standard form as ax2 + bx + c = 0, where a, b, c are constants and x is the variable. The values of x that satisfy the equation are called solutions of the equation, and a quadratic equation has at most two solutions.
Given here: The expression x²+4x-12=0
Thus simplifying the equation we get
x²+2.2x+4-12-4=0
(x+2)²-4²=0
(x+6) (x-2)=0
x=-6,2
or the equation can also be rewritten as (x+2)²=16
Thus the equivalent equation is (x+2)²=16
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a circle is centered at (-8, -13) and has a radius of 13. what is the equation of the circle? enter the equation in the box using lower case variables x and y.
Answer:
(x+8)² + (y+13)² = 169
Step-by-step explanation:
Equation of circle:
(x-h)² + (y-k)² = r²
(h,k) is the centre, r is the radius
(x-(-8))² + (y-(-13))² = 13²
(x+8)² + (y+13)² = 169
Answer: the equation of the circle is
(x + 8)² + (y + 13)² = 169
Step-by-step explanation:
A circle is the set of all points in a plane equidistant from a fixed point called the origin or center.
The center of the circle is (-8, -13)
The formula for determining the equation of a circle us expressed as
(x - h)² + (y - k)² = r²
Where
r represents the radius of the circle
h and k represents the x and y coordinates of the center of the circle. Comparing with the given points,
h = - 8 and k = - 13
Radius, r = 13
Substituting into the formula, it becomes
(x - h)² + (y - k)² = r²
(x - - 8)² + (y - - 13)² = 13²
(x + 8)² + (y + 13)² = 169
The frequency f of vibration of a violin string is inversely proportional to its length L. The constant of proportionality k is positive and depends on the tension and density of the string.
(A) write an equation that represents this variation.
(B) what effect does doubling the length on the string have on the frequency of its vibration?
Answer:
(A) [tex]f=\frac{k}{L}[/tex]
(B) Frequency becomes half.
Step-by-step explanation:
We have been given that the frequency f of vibration of a violin string is inversely proportional to its length L. The constant of proportionality k is positive and depends on the tension and density of the string.
(A) We know that two inversely proportional quantities are in form [tex]y=\frac{k}{x}[/tex], where y is inversely proportional to x and k is constant of proportionality.
Upon substituting our given values, we will get:
[tex]f=\frac{k}{L}[/tex]
Therefore, our required equation would be [tex]f=\frac{k}{L}[/tex].
(B) For part, we have been given that length is twice, so our new frequency will be [tex]f_n[/tex] and new length [tex]L_n[/tex] is [tex]2L[/tex].
Upon substituting [tex]2L[/tex] in our equation as:
[tex]f_n=\frac{k}{L_n}[/tex]
[tex]f_n=\frac{k}{2L}[/tex]
[tex]f_n=\frac{1}{2}\cdot \frac{k}{L}[/tex]
[tex]f_n=\frac{1}{2}\cdot f[/tex]
Upon comparing [tex]f_n[/tex] with [tex]f[/tex], we can see that [tex]f_n[/tex] is half the value of [tex]f[/tex].
Therefore, the frequency of vibration of violin gets half, when we double the length of the string.
The frequency f of a vibrating string is represented by the equation f = k/L, where k is a constant and L is the string's length. If the length of the string is doubled, the frequency of the string's vibration is halved due to the inverse relationship.
Explanation:Answer:(A) Considering the problem describes the frequency f of a vibrating string being inversely proportional to its length L, we can represent this information mathematically with the equation f = k/L, where k is a constant of proportionality. This constant is positive and depends on the tension and density of the string.
(B) Because frequency and length are inversely proportional, if you double the length of the string (L), the frequency (f) will halve. This is due to the inverse relationship, with the longer string taking more time to complete each vibration and thus reducing the frequency of vibration.
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: Let x represent any number in the set of odd integers less than 5. Which inequality is true for all values of x? A x<0 B x>0 C x<5 D x>5
Answer:
(C) X<5
Step-by-step explanation:
Set of all old integers less than 5=(1,3)
X will represent this in inequality :
X<5.
Therefore, the answer is C.
what is the segment of AD?
A) 5
B) 6
C) 4.5
D) 3
Answer:
i am pretty sure it is B
Step-by-step explanation:
What percent of 7.5 is 0.9
Answer:
12%
Step-by-step explanation:
0.9/7.5 *100
= 12%
Scientific research on popular beverages consisted of 70 studies that were fully sponsored by the food industry, and 30 studies that were conducted with no corporate ties. Of those that were fully sponsored by the food industry, 15 % of the participants found the products unfavorable, 23 % were neutral, and 62 % found the products favorable. Of those that had no industry funding, 38 % found the products unfavorable, 16 % were neutral, and 46 % found the products favorable.a. What is the probability that a participant selected at random found the products favorable?
b. If a randomly selected participant found the product favorable, what is the probability that the study was sponsored by the food industry?
c. If a randomly selected participant found the product unfavorable, what is the probability that the study had no industry funding?
Answer:
a. 0.26
b. 0.7
c. 0.38
Step-by-step explanation:
For a)
For sponsored studies: 62 % found the products favorable. The percentage is 0.62.
For non-sponsored studies 46% found the products favorable. The percentage is 0.46
Total probability = P(A) × P (B)
= 0.46 × 0.62
= 0.2604
For b)
Probability for the food industry = 0.7
For c) 1 - ( 0.62) = 0.38
The overall probability that a participant finds the product favorable is 57.2%. If the product was found favorable, the probability the study was industry-sponsored is 75.8%. If the product was found unfavorable, the probability the study was not industry-funded is 71.7%.
Explanation:The first step to answer the given question is understand that this is a problem of conditional probability, related to the concept of favorability or unfavorability towards products based on industry-sponsored studies and non-sponsored studies.
To find the overall probability of a participant finding the product favorable, we add the probabilities of a product being favorable under both industry funding and no industry funding, weighted by their respective chances of occurrence.
Begin by finding probability that industry sponsored and product was favorable: ((70/100) * (62/100) = 0.434)Then find probability that no industry funding and product was favorable: ((30/100) * (46/100) = 0.138)Add these two probabilities together, which gives 0.434 + 0.138 = 0.572, or 57.2%To work out the conditional probabilities:
For an industry sponsored study given that the product was favorable, the probability is ((70/100) * (62/100)) divided by 0.572 = 0.758, or 75.8%For the study having no industry funding given that the product was unfavorable, the probability is ((30/100) * (38/100)) divided by ((70/100)*(15/100) + (30/100)*(38/100)) = 0.717, or 71.7%Learn more about Probability here:https://brainly.com/question/22962752
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Sandra earned $8,000.00 from a summer job and put it in a savings account that earns 3% interest compounded continuously. When Sandra started college, she had $8,327.00 in the account which she used to pay for tuition. How long was the money in the account? Round your answer to the nearest month.
____ years and ____ months
Answer: 1 year and 4 months
Step-by-step explanation:
The formula for continuously compounded interest is
A = P x e (r x t)
Where
A represents the future value of the investment after t years.
P represents the present value or initial amount invested
r represents the interest rate
t represents the time in years for which the investment was made.
e is the mathematical constant approximated as 2.7183.
From the information given,
P = 8000
r = 3% = 3/100 = 0.03
A = 8327
Therefore,
8327 = 8000 x 2.7183^(0.03 x t)
8327/8000 = 2.7183^(0.03t)
1.040875 = 2.7183^(0.03t)
Taking log of both sides, it becomes
Log 1.040875 = log 2.7183^(0.03t)
0.0174 = 0.03tlog2.7183
0.0174 = 0.03t × 0.434 = 0.01302t
t = 0.0174/0.01302
t = 1.336
0.336 × 12 = 4.032
Therefore, the money waa in the account for 1 year and 4 months
The amount of money in Tara's account with respect to the day of the month was recorded for 31 days. The correlation coefficient was calculated to be r = −0.9870. Interpret the meaning of the correlation coefficient in terms of the scenario.
There is a weak, negative correlation between the amount of money in Tara's account and the day of the month.
There is a strong, positive correlation between the day of the month and the amount of money in Tara's account.
There is a strong, negative correlation between the amount of money in Tara's account and the day of the month.
There is a weak, positive correlation between the day of the month and the amount of money in Tara's account.
There is no correlation between the amount of money in Tara's account and the day of the month.
Answer:
There is a strong, negative correlation between the amount of money in Tara's account and the day of the month.
Step-by-step explanation:
r is between -1 and +1. r values close to -1 are strongly negative. r values close to +1 are strongly positive. r values close to 0 are weak.
r = -0.9870 is strongly negative. So there is a strong, negative correlation between the amount of money in Tara's account and the day of the month.
Final answer:
The correlation coefficient r = -0.9870 shows a strong, negative correlation between the day of the month and the amount of money in Tara's account, meaning as the days pass, her account balance generally decreases. So, the correct statement is 'There is a strong, negative correlation between the amount of money in Tara's account and the day of the month.'
Explanation:
The correlation coefficient, denoted as r, measures the strength and direction of the linear relationship between two variables. Given that the correlation coefficient in the scenario is r = −0.9870, this indicates a strong, negative correlation between the amount of money in Tara's account and the day of the month. This means that as the days of the month increase, the amount of money in Tara's account tends to decrease, and this pattern is quite consistent, considering the high absolute value of the coefficient, which is close to -1.
Students are getting signatures for a petition to increase sports activities at the community center. The number of signatures they get each day is 3 times as many as the day before. The expression 3 to the 6th power represents the number of signatures they got on the sixth day. How many signatures did they get on the first day?
Answer:
Students got 3 signature on the first day.
Step-by-step explanation:
We are given the following in the question:
The number of signatures they get each day is 3 times as many as the day before.
Thus, the number of signature forms a G.P with common ratio, r = 3.
The expression 3 to the 6th power represents the number of signatures they got on the sixth day. Thus we can write:
[tex]a_6 = 3^6[/tex]
The [tex]n^{th}[/tex] term in a G.P is given by
[tex]a_n = a_1r^{n-1}[/tex]
where [tex]a_1[/tex] is the first term in the G.p
Putting values, we get,
[tex]3^6 = a_1(3)^{6-1}\\3^6 = a_1(3)^5\\\Rightarrow a_1 = 3[/tex]
Thus, the first term of G.P is 3.
Hence, students got 3 signature on the first day.
A square is 3 inches on each side. A small square, x inches on each side, is cut out from each corner of the original square. Represent the area of the remaining portion of the square in the form of a polynomial function A(x)
Here's the polynomial function representing the area of the remaining portion of the square:
A(x) = 9 - 4x + x^2
1. **Initial area:** The original square has a side length of 3 inches, so its initial area is 3 * 3 = 9 square inches.
2. **Removing squares:** When small squares of side length x are cut out from each corner, the remaining shape becomes a smaller square with a side length of (3 - 2x) inches.
3. **New area:** The area of this smaller square is (3 - 2x) * (3 - 2x) = 9 - 6x + 4x^2 = 4x^2 - 6x + 9.
4. **Simplifying:** We can rearrange this expression to get a simpler polynomial function: A(x) = 9 - 4x + x^2.
Therefore, A(x) = 9 - 4x + x^2 represents the area of the remaining portion of the square after the small squares are cut out, as a function of the side length x of the removed squares.
Peter is Distributing pamphlets about dog care and samples of dog biscuits. The dog biscuits come in packages of 12 in the pamphlets are in packages of 20 how many packages of dog biscuits and pamphlets will Peter need
Answer:
3 packeges of Pamphlets and 5 packages of dog biscuits.
Step-by-step explanation:
Peter have to distribute a pamphlet with one dog buscuit to each person they need equal amount of pamphlets and biscuitsby taking LCM of both 12 and 20 theanswer is 60. Peter needs 60 pamphlets and 60 dog biscuits. So he require 3packages of pamphlets and 5packages of dog biscuits.
FUNCTIONS: In the space provided, type the answer in descending order as it applies without any spaces between the letters, numbers, or symbols.
Type the composition (fog)(x) of the given functions:
f(x) = x^2 + 2x − 6 and g(x) = x + 5.
Answer:
Hence The composition [tex](fog)(x)[/tex] of the given function is [tex]x^2+12x+29[/tex].
Step-by-step explanation:
Given:
[tex]f(x) = x^2+2x-6[/tex]
[tex]g(x)=x+5[/tex]
We need to find [tex](f o g)(x)[/tex].
Solution:
Now we can say that;
[tex](f o g)(x)[/tex] = [tex]f(g(x))[/tex]
[tex](fog)(x) = (x+5)^2+2(x+5)-6[/tex]
Now Applying distributive property we get;
[tex](fog)(x) = (x+5)^2+2\times x+2\times5-6\\\\(fog)(x) = (x+5)^2+2x+10-6\\\\(fog)(x) = (x+5)^2+2x+4[/tex]
Now Solving the exponent function we get;
[tex](fog)(x) = x^2+2\times x\times 5+5^2+2x+4\\\\(fog)(x) = x^2+10x+25+2x+4\\\\(fog)(x) = x^2+12x+29[/tex]
Hence The composition [tex](fog)(x)[/tex] of the given function is [tex]x^2+12x+29[/tex].
Maggie's mom agrees to let Maggie buy small gifts for some friends. Each gift costs $4. Maggie's mom gave her a budget of $19. When Maggie went online to order gifts, she discovered there was a $7 shipping fee no matter how many gifts she bought.
Answer:
If you`re trying to figure out how many gifts Maggie can get without the fee going over the budget, then I would say that she can buy 3 gifts. 4 times 3 is 12, and when you add 7 it equals 19. Of course, if 3 is pushing it, Maggie can also buy 1 or 2 gifts, and she`d be fine. Hope this helps!
Step-by-step explanation:
Three brothers Charlie Robert and Nicholas have 21 video games Charlie has twice as many games as Robert. Robert has 5 fewer games than Nicholas what would be the equation
Answer:
The equation would be [tex]4x-15=21[/tex].
Step-by-step explanation:
Given;
Total Number of Video games = 21
Let the number of video game Nicholas has be 'x'.
Now Given:
Robert has 5 fewer games than Nicholas.
so we can say that;
number of video game Robert has = [tex]x-5[/tex]
Also Given:
Charlie has twice as many games as Robert.
so we can say that;
number of video game Charlie has = [tex]2(x-5)=2x-10[/tex]
we need to write the equation.
Solution:
Now we can say that;
Total Number of Video games is equal to sum of number of video game Nicholas has, number of video game Robert has and number of video game Charlie has.
framing the equation we get;
[tex]x+x-5+2x-10=21\\\\4x-15=21[/tex]
Hence The equation would be [tex]4x-15=21[/tex].
On Solving we get;
Adding both side by 15 we get;
[tex]4x-15+15=21+15\\\\4x=36[/tex]
Dividing both side by 4 we get;
[tex]\frac{3x}{4}=\frac{36}{4}\\\\x=9[/tex]
Hence Nicholas has = 9 video games
Robert has = [tex]x-5= 9-5=4[/tex] video games
Charlie has = [tex]2x-10 = 2\times9-10=18-10=8[/tex] video games
Answer:
The equation would be [tex]4x-15=21.[/tex]
Step-by-step explanation:
Given:
Three brothers Charlie Robert and Nicholas have 21 video games Charlie has twice as many games as Robert. Robert has 5 fewer games than Nicholas.
Now, to find the equation.
Let the games Nicholas has be [tex]x.[/tex]
Robert has games [tex]x-5.[/tex]
And Charlie has [tex]2(x-5).[/tex]
Now, the equation is:
[tex](x)+(x-5)+2(x-5)=21.[/tex]
[tex]x+x-5+2x-10=21\\2x-5+2x-10=21\\4x-15=21[/tex]
[tex]4x-15=21.[/tex]
Therefore, the equation would be [tex]4x-15=21.[/tex]
Carrie earned $3673 from a summer job and put it in a savings account that earns 10% interest compounded annually. When Carrie started college, she had $7614 in the account which she used to pay her tuition. How long was the money in the account?
Answer:
Step-by-step explanation:
We would apply the formula for determining compound interest which is expressed as
A = P(1+r/n)^nt
Where
A = total amount in the account at the end of t years
r represents the interest rate.
n represents the periodic interval at which it was compounded.
P represents the principal or initial amount deposited
From the information given,
A = 7614
r = 10% = 10/100 = 0.1
n = 1 because it was compounded once in a year.
P = 3673
Therefore,
7614 = 3673(1+0.1/1)^1 × t
7614/3673 = 1.01^t
2.073 = 1.01^t
Taking log of both sides, it becomes
Log 2.073 = log 1.01^t
0.3166 = t × 0.0043
t = 0.3166/0.0043
t = 73.3
Suppose Paul went to the store and bought 4 peaches to add to the basket.Write two new numerical expressions to represent the total number of fruit in the basket.
The question involves creating numerical expressions to represent the total number of fruits in a basket after adding 4 peaches. By assuming the initial fruit count as x, two expressions are x + 4, representing the total number of fruits, and 2(x + 4), indicating a scenario where the total fruit count doubles after adding the peaches.
Explanation:The question asks for two new numerical expressions to represent the total number of fruit in the basket, given that Paul added 4 peaches. To create these expressions, we need to assume there was an initial number of fruit in the basket before Paul added the peaches. Let's denote the initial number of fruit as x. Therefore, our two new numerical expressions could be:
x + 4: This expression represents the total number of fruit in the basket after adding 4 peaches to the initial amount.2(x + 4): This expression might represent a scenario where, for some reason, the number of fruits, after adding the 4 peaches, is doubled. It exemplifies how numerical expressions can model different real-world scenarios beyond simple addition.These examples show how basic algebraic expressions can be used to represent situations involving changes in quantity.
Translate the given statement into a linear equation in the form ax + by = c using the indicated variable names. Do not try to solve the resulting equation. HINT [See Example 7 and the end of section FAQ.] The number of new clients (x) is 154% of the number of old clients (y).
The statement 'the number of new clients (x) is 154% of the number of old clients (y)' can be translated into a linear equation in the standard form 'ax + by = c' as '-1.54y + x = 0' or '-154y + 100x = 0' where a, b, and c are constants.
Explanation:To translate the given statement into a linear equation, we need to interpret the percentages as a ratio. When we say 'the number of new clients (x) is 154% of the number of old clients (y)', it means 'x is equivalent to 1.54 times y'. So, we can write that as a linear equation:
x = 1.54y
However, that's not in the form 'ax + by = c'. To get it into that form, we can write it as:
-1.54y + x = 0
Or alternately you might prefer, if you wish to avoid decimals:
-154y + 100x = 0
Where a = -154, b = 100 and c = 0.
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A ball is launched from 8 yards off the ground and travels in parabolic motion, landing in a net 80 yards away and also 8 yards off the ground. A wall lies 75% of the way down the path, which the ball cleared by 13 yards.
If the ball’s maximum height during its flight was 80 yards, how many yards tall is the wall?
Answer:
The answer to this question is the wall is 45.95 yards tall
Step-by-step explanation:
To solve this, we list out the given variables and the unknowns thus
Height of ball at launch = 8 yards
Distance of net from the ball = 80 yards
Distance of the wall down the path = 75%
Maximum height of the ball= 80 yards
equation of Motion of the ball = parabolic motion =
v² = u² - 2gS
S = 80 - 8 = 72 yards
at maximum height v = 0 thus u² = 2×9.81×72 =1412.64
u = 37.59 m/s
also v = u - gt and again at max height v = 0
Therefore 37.59 = 9.81×t or t = 3.83 s
If the motion of the ball is free of obstruction then time of flight before the ball just reaches the 8 yards off the ground = 3.83×2 = 7.66 seconds
Taking the initial velocity as zero at maximum height and from the equation
S = ut + 0.5×gt² we get, where S is the heigt of the ball from touching the actual field ground which is 80 yards we have
80 = 0.5×9.81×t²
so that t² = 2×80÷9.81 = 16.31 or t = 4.04s
Therefore the total time of flight = 4.04 + 3.83 = 7.87 seconds
if the ball is considered as having a constant horizontal velocity, therefore
at 75% of the way the time it took will be 0.75×7.87 = 5.9 seconds
However time it took the ball to reach maximum height and then starts descent = 3.83s, and the time at which the ball is directly over the wall = 2.07 seconds on the second half just after reaching mximum height
Thus at 2.07 seconds the distance trvelled from the maximum height is
S = ut +0.5gt² as before where u = 0
hence S = 0.5×9.81×2.07² = 21.05 yards or (80 -21.05) yards off the ground = 58.95 yards
As stated in the question, the ball cleared the wall by 13 yards therefore the height of the wall is 58.95 - 13 = 45.95 yards
Janine is saving money to buy a car. She has a total of $1400 left to save, and she plans to save a certain percentage of $1400 each month: In August, Janine will save 25% of $1400. In September, Janine will save 40% of $1400. In October, Janine will save 15% of $1400. In November, Janine will save the remaining amount. Which option correctly explains what Janine plans to save each month?
Answer:
August=$350
September=$560
October=$210
November= $280
Step-by-step explanation:
Total amount left to save $1400.
We calculate the amount save each mount since we are already given the percentages.
For the month of August she saves 25% of $1400, we convert the percentage to equivalent cash
[tex]\frac{25}{100}*1400\\=350[/tex]
for the month of August, she saved $350,
Next For the month of September she saves 40% of $1400, we convert the percentage to equivalent cash
[tex]\frac{40}{100}*1400\\=560[/tex]
for the month of September, she saved $560,
Next For the month of October she saves 15% of $1400, we convert the percentage to equivalent cash
[tex]\frac{15}{100}*1400\\=210[/tex]
for the month of October, she saved $210,
total amount saved by October = $350+$560+$210=$1120
Amount saved by November=1400-1120=$280
Manny is an online student who currently owns an older car that is fully paid for. He drives, on average, 110 miles per week to commute to work. With gas prices currently at $2.65 per gallon, he is considering buying a more fuel-efficient car, and wants to know if it would be a good financial decision. The old car Manny owns currently gets 16 miles per gallon for average fuel efficiency. It has been a great vehicle, but with its age, it needs repairs and maintenance that average $740 per year (as long as nothing serious goes wrong). The newer, more fuel-efficient car that he is looking at to purchase will cost a total of $6,500 over a three-year loan process. This car gets 28 miles per gallon and would only require an average of $10 per month for general maintenance. To help make a decision, Manny wants to calculate the total cost for each scenario over three years. He decides to use the quantitative reasoning process to do this.
In this exercise we have to use the knowledge of finance to identify the best cost benefit is to buy a new car or a used car, thus we find that:
The new car costs more, that can be prove if;
Old car: [tex]\$5062.125[/tex] New car: [tex]\$7161.357[/tex]
Manny drives an average of 110 miles per week with his old car. The old car gets 16 miles per gallon. The cost per gallon is [tex]\$2.65[/tex] repair and maintainance costs an average of [tex]\$740[/tex] per year. For the old car, to find the amount spent on the car we have:
[tex](110/16) * (2.65) = \$18.21875 / week[/tex]
There are 52 weeks in a year. We have:
[tex](10.21875)*(52) = \$ 947.375\\947.375 + 740 = \$1687.375\\(1687.357) * (3) = \$5062.125[/tex]
The new car cost [tex]\$6500[/tex] over a three year loan process. The car gets 28 miles per gallon. It requires a maintenance of [tex]\$10[/tex] per month. For the new car to find the amount, we have:
[tex](110)*(28) * (2.65) *(52) = \$541.357\\541.357 + 10*(12) + 6500 = \$ 7161.357[/tex]
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