Answer:
38.9 square inches.
Step-by-step explanation:
We are asked to find the difference in area covered by a single 3 inch windshield wiper operating with a central angle of 138 degrees compared to a pair of 5 inch wipers operating together each having a central angle of 114 degrees.
We will use area of sector formula to solve our given problem as:
[tex]\text{Area of sector}=\frac{\theta}{360}\times \pi r^2[/tex], where, r represents radius and theta represents central angle.
Let us find area of sector with central angle 140 degree and radius 3 inch.
[tex]\text{Area of sector}=\frac{138}{360}\times \pi (3)^2[/tex]
[tex]\text{Area of sector}=\frac{138}{360}\times 9\pi[/tex]
[tex]\text{Area of sector}=10.83849[/tex]
Now, we will find area of sector with central angle 114 degree and radius 5 inch and multiply by 2 as:
[tex]\text{Area of sector}=2(\frac{114}{360}\times \pi (5)^2)[/tex]
[tex]\text{Area of sector}=2(\frac{114}{360}\times 25\pi)[/tex]
[tex]\text{Area of sector}=\frac{114}{360}\times 50\pi[/tex]
[tex]\text{Area of sector}=49.74188[/tex]
Let us find difference of area as shown below:
[tex]\text{Difference of areas}=49.74188-10.83849[/tex]
[tex]\text{Difference of areas}=38.90339[/tex]
[tex]\text{Difference of areas}\approx 38.9[/tex]
Therefore, the difference in area covered is approximately 38.9 square inches.
Final answer:
Calculate the areas covered by differently sized and angled windshield wipers to find the difference.
Explanation:
Single 3-inch wiper:
Area covered = (1/2) × r² × (2 × pi × (angle/360))Pair of 5-inch wipers:
Each wiper's area = (1/2) × r² × (2 × pi × (angle/360))Total area covered by the pair = 2 × (area of a single wiper)The difference in area cleaned by the pair of 5-inch wipers compared to the single 3-inch wiper is:
A-diff = A-combined - A-single ≈ 31.50 sq in - 7.07 sq in ≈ 24.43 sq in
Therefore, the pair of 5-inch wipers cleans approximately 24.43 square inches more area than the single 3-inch wiper.
Suppose the age of people in a certain population are distributed normally with a mean of 37.5 years and standard deviation of 6.2 years. What is the probability of randomly selecting a person who is over 45 years old given that they are older than 40.
Answer:
[tex] P(X>45 | X>40)= \frac{P(X>45 \cap X>40)}{P(X>40)}= \frac{P(X>45)}{P(X>40)}= \frac{0.113}{0.343}= 0.329[/tex]
Step-by-step explanation:
Previous concepts
Normal distribution, is a "probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean".
The Z-score is "a numerical measurement used in statistics of a value's relationship to the mean (average) of a group of values, measured in terms of standard deviations from the mean".
Solution to the problem
Let X the random variable that represent the age of a population, and for this case we know the distribution for X is given by:
[tex]X \sim N(37.5,6.2)[/tex]
Where [tex]\mu=37.5[/tex] and [tex]\sigma=6.2[/tex]
We are interested on this probability:
[tex] P(X>45 | X>40)= \frac{P(X>45 \cap X>40)}{P(X>40)}= \frac{P(X>45)}{P(X>40)}[/tex]
We can begin finding [tex] P(X>40)[/tex] using the z score formula given by:
[tex] z = \frac{a-\mu}{\sigma}[/tex]
Using this formula we have:
[tex] P(X>40)= P(Z>\frac{40-37.5}{6.2}) = P(Z>0.403)[/tex]
And using the complement rule and the normal standard table or excel we have this:
[tex]P(Z>0.403)=1-P(Z<0.403)= 1-0.657= 0.343[/tex]
Now we can find [tex] P(X>45)[/tex] using the z score formula given by:
[tex] z = \frac{a-\mu}{\sigma}[/tex]
Using this formula we have:
[tex] P(X>45)= P(Z>\frac{45-37.5}{6.2}) = P(Z>1.210)[/tex]
And using the complement rule and the normal standard table or excel we have this:
[tex]P(Z>1.210)=1-P(Z<1.210)= 1-0.887= 0.113[/tex]
And replacing into our original probability we got:
[tex] P(X>45 | X>40)= \frac{P(X>45 \cap X>40)}{P(X>40)}= \frac{P(X>45)}{P(X>40)}= \frac{0.113}{0.343}= 0.329[/tex]
Paul is six years older than Gary.
The sum of their ages is 30.
What are their ages?
Answer: Paul is 18 years old and Gary is 12 years old.
Step-by-step explanation:
Let x represent the age of Paul.
Paul is six years older than Gary. This means that Gary's age would be
x - 6
The sum of their ages is 30. This means that
x + x - 6 = 30
2x - 6 = 30
Adding 6 to the left hand side and the right hand side of the equation, it becomes
2x - 6 + 6 = 30 + 6
2x = 36
Dividing the left hand side and the right hand side of the equation by 2, it becomes
2x/2 = 36/2
x = 18
Therefore, Paul is 18 years old.
Gary is 18 - 6 = 12 years old.
A paper company needs to ship paper to a large printing business. The paper will be shipped in small boxes and large boxes. Each small box of paper weighs 40 pounds and each large box of paper weighs 70 pounds. A total of 20 boxes of paper were shipped weighing 1220 pounds altogether. Write a system of equations that could be used to determine the number of small boxes shipped and the number of large boxes shipped. Define the variables that you use to write the system.
Answer: The system of equations required is
x + y = 20
40x + 70y = 1220
Step-by-step explanation:
Let x represent the number of small boxes of paper shipped and
Let y represent the number of large boxes of paper shipped.
A total of 20 boxes of paper were shipped. This is expressed as
x + y = 20
Each small box of paper weighs 40 pounds and each large box of paper weighs 70 pounds. The total weight of the large and small boxes of paper that were shipped is 1220 pounds altogether. This is expressed as
40x + 70y = 1220
We define x as the number of small boxes and y as the number of large boxes. The first equation, x + y = 20, is based on the total number of boxes. The second equation, 40x + 70y = 1220, is based on the total weight of the boxes.
Explanation:We need to find a system of equations that represents the given situation. We will use two variables. Let's say x represents the number of small boxes and y represents the number of large boxes for your problem.
The first equation can be based on the total number of boxes, which is 20. So, the equation is: x + y = 20.
The second equation will be based on the total weight of the boxes, which is 1220 pounds. A small box weighs 40 pounds and a large box weighs 70 pounds. So, the equation will be: 40x + 70y = 1220.
So, we have the system of equations:
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A yard of lace costs w cents a yard and fabric costs $.40 more than the lace. Kimberly wants to buy one yard of lace and 2 yards of fabric. Mow much money will she need? Express your answer in terms of w.
Answer:
Step-by-step explanation:
A yard of lace costs w cents a yard and fabric costs $.40 more than the lace. This means that the cost of a yard of fabric would be
w + 0.5
Kimberly wants to buy one yard of lace and 2 yards of fabric. This means that the total amount of money that she would have to pay for the lace is is
0.4 × 1 = 0.40
Amount that she would spend on the fabric is 2(w + 0.5) = 2w + 1
Total cost would be
0.4 + 2w + 1
Final answer:
Kimberly will need a total of 3w + 80 cents to purchase one yard of lace and two yards of fabric, where w is the cost of one yard of lace in cents.
Explanation:
To calculate how much money Kimberly will need to buy one yard of lace and two yards of fabric, first we identify the cost of one yard of lace as w cents. The fabric costs $0.40 more than the lace per yard, so the cost of one yard of fabric is w cents + 40 cents. Kimberly wants to buy two yards of fabric, so we multiply the cost of one yard of fabric by 2, which gives us 2(w + 40) cents.
Adding together the cost of the lace and the two yards of fabric, we get: w + 2(w + 40). Simplifying this expression, we have: w (cost of one yard of lace) + 2w (twice the cost of lace per yard for two yards of fabric) + 80 (twice the additional cost of fabric per yard)
w + 2w + 80 cents
3w + 80 cents
Therefore, Kimberly will need a total of 3w + 80 cents to purchase one yard of lace and two yards of fabric.
Rosa earns $120 per week tutoring math. Each week, she puts $36 from her paycheck in her bank account to save for college. Rosa wants to know what percent of her earnings she saves.
Final answer:
Rosa saves 30% of her weekly earnings for college, which is calculated by dividing the amount saved ($36) by her total weekly earnings ($120) and then multiplying by 100%.
Explanation:
To determine the percentage of her earnings that Rosa saves, we will use the formula for calculating percentage: Percentage saved = (Amount saved ÷ Total earnings) × 100%.
First, we identify the total earnings and the amount saved: Rosa's total weekly earnings are $120, and she saves $36 each week.
Next, we calculate the percentage: Percentage saved = ($36 ÷ $120) × 100% = 0.3 × 100% = 30%.
Therefore, Rosa saves 30% of her weekly earnings for college.
Hank and debra each own two milking cows. One day, they milked their cows and compared the amount of milk the cows prodyce in that one day. How many more gallons of milk did debras two cowsbprodyce on that day compared to hanls two cows?
Debra's cows produced [tex] 2 \frac{7}{24}[/tex] more gallons than Hank's cows.
Hank's cows :
4¾ + 4⅛ = 8⅞Debra's Cows :
5½ + 5⅔ = 11⅙The difference in amount of Milk produced :
Sum of Debra's cow - Sum of Hank's cows
Now we have:
11⅙ - 8⅞
67/6 - 71/8 = (536 - 426) / 48
67/6 - 71/8 = 110/48
110/48 = [tex] 2 \frac{7}{24}[/tex]
Hence, Debra's cows produced [tex] 2 \frac{7}{24}[/tex] more gallons than Hank's cows.
The height of Mountain P is 1,086 feet.The height of Mountain Q is 4 times the height of Mountain P.The area model shown below represents one way to find the height of Mountain Q. What are missing values for a,b,c in the model?
Answer:
A=4000, B=80, C=24
Step-by-step explanation:
You forgot to put the correct area model, I attached it to the answer.
We have the fact that Mountain Q is 4 times the height of Mountain P. That's the "4" we have in the left side of our model. It's like having a multiplication table, next to the "4" we have "A" and upper this we have "1000", the only thing we have to do is multiplify 4*1000=4000. The next letter we have is B and below it we have "320", we divided it by 4, 320/4=80. The last letter we have is C, and is below a "6", we only have to multiplify it by 4, 6*4=24.
At the end we only sum our
A + 320 + c = 4344 (4 times the height of Mountain P).1000 + B + 6 = 1086(the height of the Mountain P).The missing values in the model for the area are:
A = 4,000
B = 80
C = 24
What is a Model?A model is a mathematical system that represents a real life concept in an easy to understand manner.
Given the model in this question, we can find the missing values as shown below:
A = 4 × 1000 = 4,000
B = 320/4 = 80
C = 4 × 6 = 24
Therefore, the missing values in the model for the area are:
A = 4,000
B = 80
C = 24
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A taxi cab costs $1.75 for the first mile and $0.75 for each additional mile. You have $20 to spend on your ride. Which inequality could be solved to find how many miles you can travel, if n is the number of additional miles? A 1.75n + 0.75 ≤ 20 B 1.75n + 0.75 ≥ 20 C 0.75n + 1.75 ≥ 20 D 0.75n + 1.75 ≤ 2
n = the number of miles traveled
0.75n + 1.75 ≤ 20 Option D
[costs $0.75 for each additional mile (n) plus $1.75 for the first mile less than or equal to $20 (because you have a maximum of $20 and you can't spend more than that)]
The inequality to determine the number of additional miles you can travel with $20, considering a base fare of $1.75 and $0.75 per additional mile, is 0.75n + 1.75 ≤ 20.
Explanation:The subject of this question is to determine which inequality represents the situation of a taxi cab fare and how many miles can be traveled with $20 given the cost per mile. The correct inequality is the one that starts with the base fare and adds the cost for each additional mile multiplied by the number of miles. The base fare is $1.75 for the first mile and there is an additional cost of $0.75 for each additional mile.
Let n represent the number of additional miles you can travel. Since you have $20, the inequality to find out how many miles you can travel would be the money spent on additional miles plus the base fare has to be less than or equal to $20 you have. This is represented by the inequality 0.75n + 1.75 ≤ 20. This means we are adding $0.75 for each additional mile (n) to the base fare of $1.75, and the total cost must not exceed $20 to stay within the budget.
In a lilac paint mixture 40% of the mixture is white paint 20% is blue and the rest is red there are four cups of blue paint used in a batch of lilac paint how many cups of white paint is used
Answer: 8 cups of white paint is used.
Step-by-step explanation:
In a lilac paint mixture 40% of the mixture is white paint 20% is blue and the rest is red. This means that the percentage of red paint in the mixture is 100 - (40 + 20) = 40%
There are four cups of blue paint used in a batch of lilac paint. This means that 20% of the total number of cups of paint used in a batch of lilac paint is 4.
Assuming that the total number of cups of paint in the mixture is x, then,
20/100 × x = 4
0.2x = 4
x = 4/0.2 = 20
Therefore, the number of cups of white paint used is
40/100 × 20 = 0.4 × 20
= 8 cups
Final answer:
In the lilac paint mixture, for every 4 cups of blue paint, which accounts for 20% of the mixture, there are 8 cups of white paint, corresponding to 40% of the mixture.
Explanation:
The question involves determining the amount of white paint used in a batch of lilac paint given that 40% of the paint mixture is white, 20% is blue, and the remainder is red. We're told that 4 cups of blue paint are used. Since blue paint represents 20% of the mixture, we can use this information to find out the total amount of the paint mixture and then calculate the amount of white paint needed.
First, find the total amount of the paint mixture by calculating the full 100% that the 4 cups of blue paint (20%) contribute to. This calculation is as follows:
Total Paint = 4 cups (20%) / 0.20Total Paint = 20 cupsNow that we know the total paint mixture is 20 cups, we can determine the amount of white paint, which is 40% of the total mixture:
White Paint = Total Paint x 40%White Paint = 20 cups x 0.40White Paint = 8 cupsTherefore, 8 cups of white paint are used in the mixture.
We have three fair coins, each of which has probability 1/2 of having a heads outcome and a tails outcome. The experiment is to flip all three coins and observe the sequence of heads and tails. For example, outcome HTH means coin 1 was heads, coin 2 was tails, coin 3 was heads. Note that there are 8 total outcomes, and we assume that each one is equally likely. What is the probability that the outcome has at least two consecutive heads in the sequence?
Answer: 3/8
Step-by-step explanation:
Firstly, let's look at the possible outcome when 3 coins are tossed.
If two coins are first tossed, the possible outcome will be,
{HH, HT, TH, TT}
if one more coin is tossed together with the two to make it 3coins, the possible outcome will be gotten by matching the H and T of the third coin with the set of sample space above to give us,
S = {HHH, HHT, HTH, HTT, THH, THT, TTH, TTT}
This gives a total sample space of 8.
Outcomes that has at least two consecutive heads in the sequence are {HHH, HHT, THH} which is the possible outcome i.e 3
Probability that the outcome has at least two consecutive heads in the sequence will be;
Possible outcome/total outcome
= 3/8
The polynomial negative 8.5 x squared plus 103 x plus 2026−8.5x2+103x+2026 models the yearly number of visitors (in thousands) x years after 20062006 to a park. Use this polynomial to estimate the number of visitors to the park in 20182018.
Answer:
2038000 visitors
Step-by-step explanation:
Now in question expression has been repeated, so original expression is
[tex]-8.5x^2+103x+2026[/tex]
Now according to given information "x" represent the number of years after 2006. So, for 2018, it will be
2018-2006 = 12 years
So, x = 12
putting value of x in expression
(-8.5)x(12)^2 + (103)x(12) + 2026
-1224 + 1236 + 2026
2038
Now the model shows the number of visitors in thousand, so you have to multiply your answer in thousand
2038 x 1000
= 2038000 visitors
A grocery store has 12 cartons of yogurt for sale, of which 3 are raspberry. What is the probability that a randomly selected carton of yogurt will be raspberry?
Answer:
The answer to your question is the probability to select a yogurt of raspberry is 1/4 or 25%.
Step-by-step explanation:
Data
Number of cartons = 12
Number of cartons of raspberry = 3
To solve this problem, just use the formula of probability and simplify it to get the result.
Formula
P(A) = [tex]\frac{Number of favorable outcomes to A}{Total number of outcomes}[/tex]
Substitution
P(A) = 3/12
Simplification
P(A) = 1/4 or 25%
Elena is cutting a 30-foot piece of ribbon for a craft project. She cuts off 7 feet, and then cuts the remaining piece into 9 equal lengths of x feet each.
Answer:
The correct answer is C. 30 = 9x + 7
Correct statement and question:
Elena is cutting a 30-foot piece of ribbon for a craft project. She cuts off 7 feet, and then cuts the remaining piece into 9 equal lengths of x feet each.
Answer choices:
A. 7x + 9 = 30
B. 30x + 7 = 9
C. 30 = 9x + 7
D. 9x - 7 = 30
Source:
https://quizizz.com/admin/quiz/5c94aa300d3459001a4ef259/unit-6-lesson-4-equations-and-word-problems
Step-by-step explanation:
1. Information given to us to answer the problem correctly:
Length of the piece of ribbon for a craft project = 30 feet
First cut = 7 feet
Remaining piece cut into 9 equal lengths of x feet each
2. Let's find the right equation to solve for x:
9x + 7 = 30
The nine equal pieces of x feet each plus the piece of 7 feet add up to 30 feet
The correct answer is C. 30 = 9x + 7
The probability you'll see a falling star in the sky over the course of one hour is 0.44. What's the probability you'll see one over half an hour?
Answer:
Probability so see one falling star over half an hour is 0.25
Step-by-step explanation:
An hour can be taken as two half hours so we can write the probability to see a falling star as
(1-P)*(1-P) = 1 - 0.44
( 1 - P )² = 0.56
1 - P = [tex]\sqrt{0.56}[/tex]
P = 1 - [tex]\sqrt{0.56}[/tex]
P = 0.25
Suppose that 20% of all personal computers of a certain brand break down in the first year of operations. In an office with 10 such computers, find the probability that: a) none break down; b) exactly five break down; c) at least one breaks down; d) at most two breaks down; e) all break down.
Answer:
d
Step-by-step explanation:
The probability of computers are 20%. Therefore to determine the amount that would break out of 10 we simply just multiply by 0.2.
[tex]=10\cdot{0.2}=2[/tex]
Therefore at most two breaks down. The answer is d
Isaac predicted that advertising their business would add an additional $400 out of the $900 the brothers were adding to the equipment cost. What percent is $400 out of the additional amount they were adding?
Answer:
44.4%
Step-by-step explanation:
Additional amount they were adding to equipment cost is $900
Percentage of $900 that is $400 = $400/$900 × 100 = 44.4%
$400 is approximately 44.44% of the additional $900 being added to the equipment cost.
Explanation:To calculate what percent $400 is out of the additional $900 being added to the equipment cost, we can use the following formula:
Percent = (Part / Whole) × 100%In this case, the part is $400, which is the predicted additional profit from advertising, and the whole is the total additional amount of $900 being added to the equipment cost.
Using the formula:
Percent = ($400 / $900) × 100%Percent = 0.4444... × 100%Percent = 44.44...So, $400 is 44.44% (approximately) of the additional $900 that the brothers were planning to add to the equipment cost.
Students are selling raffle tickets for a school fundraiser. They collect $25 for every 10 raffle tickets sold. Rain equation that reflects the relationship between m and r
Answer: m= $25/10 r
Step-by-step explanation:
Let m= money
r= raffle ticket
Then according to the statement
m= $25 for 10 tickets
so 10 tickets= $25
Or the equation goes,
m= $25/10 r
On another map the distance between saugerties and kingston is 2 inches. Whar would tge distance from saugerties to catskill be on this map
The question is incomplete. The map is attached as a photo and here is the complete question:
a. What is the actual distance between Saugerties and
Kingston? ___________________________
b. Catskill is 15 miles from Saugerties. What would the
distance on the map be? ___________________________
c. On another map, the distance between Saugerties and
Kingston is 2 inches. What would the distance from Saugerties to
Catskill be on this map? __________________
Since you have asked the answer for part (c) only, here is the answer:
Answer:
The distance from Saugerties to Catskill would be 3 inches on this map.
Step-by-step explanation:
On the given map, the distance between Saugerties and Kingston is 4 inches and the scale is 1 inch = 2.5 miles.
To calculate the actual distance in miles between Saugerties and Kingston, consider the given scale:
1 inch ---------- 2.5 miles
4 inches ------ x miles
Cross multiplying:
1x = 2.5 x 4
x = 10 miles
The actual distance between Saugerties and Kingston is 10 miles.
On another map, the distance between Saugerties and Kingston is 2 inches. Which means the scale is 2 inches = 10 miles. So 1 inch = 10/2 = 5 miles
The scale on the other map is 1 inch = 5 miles.
Catskill is 15 miles from Saugerties (given in the previous part). So on the map:
1 inch -------- 5 miles
y inch ------- 15 miles
Cross multiplying:
15 = 5y
y = 15/5
y = 3 inches.
The distance from Saugerties to Catskill would be 3 inches on this map.
Find the inverse of the function.
Y= -3/x+4
Yo sup??
y=-3/x+4
cross multiply
x+4=-3/y
x=-3/y-4
f(y)=-3/y-4
or
f(x)=-3/x-4
=-4x-3/x
The correct answer is option 4
Hope this helps.
Anyone know the answer for this problem??
Answer:
20%.
Step-by-step explanation:
I = PRT
300 = 3000*R * 0.5
300 = 1500R
R = 300 / 1500
R = 3/15 = 1/5
As a percentage it is 100 * 1/5
= 20%.
Answer: 20%
Step-by-step explanation: Notice that in this problem we're asked to find the interest rate not the interest so our answer will be a percent rate of interest.
Let's start this problem by writing the interest formula shown below.
Interest = principal · rate · time
Now we fill in the formula.
The interest earned is $300 so we substitute 300 into the formula.
The principal is the amount invested or $3,000.
We don't know the rate so we can use the variable r.
The time is 0.5 which is equivalent to 1/2
so we can substitute 1/2 in for t.
So we have the equation 300 = (3,000)(r)(1/2).
Simplifying on the right side of the equation, 3,000 · 1/2 is 1,500.
So we have 300 = 1,500r
Now to solve for r since r is being multiplied by 1,500, we need to divide by 1,500 on both side of the equation.
On the right side of the equation, the 1,500's cancel and we have r and on the left side of the equation we have 300 divided by 1,500 or 0.2.
This means that 0.2 = r.
Remember however that the interest rate is a percent so we need to change our decimal to a percent by moving the decimal point 2 places to the right to get 20%. So the interest rate is 20%.
My sister was fooling around wi her money the other night and left the coins in a pattern of four rows with four coins in each row. Each row had exactly one penny, one nickel,one dime , and one quater
Answer:
one penny = S
one nickel = T
one dime = U
one quater = V
∴ Probability one obtaining at least one of S, T, U or V = 4/24 = 1/6
Step-by-step explanation: They can be arranged as follows
1) STUV
2)TSUV
3) TSVU
4) UVST
5) VUST
6) VUTS
7) TVUS
8) TVSU
9) TUVS
10) SVUT
11) VSTU
12) VSUT
13) SVTU
14) UTSV
15) UTVS
16) USTV
17) USVT
18) VTUS
19) VTSU
20) SUTV
21) SUVT
22) STVU
23) TUSV
24) UVTS
Country A has twice as many workers as Country B. Country A also has twice as much physical capital, twice as much human capital, and access to twice as many natural resources as Country B. Assuming constant-returns to scale, which of the following is higher in Country A?
a. Both output per worker and productivity.
b. Output per worker but not productivity.
c. Productivity but not output per worker.
d. Neither output per worker nor productivity.
Answer: d
Step-by-step explanation: Both country A and B has equal number of workers,physical capital, human capital and access to natural resources.
Non is higher than the other.
-----40 POINTS--------
Provide the missing reasons for the proof of part of the triangle midsegment theorem.
Proof
Provide the missing reasons for the proof of part of the triangle midsegment theorem.
Given: K is the midpoint of MJ.
L is the midpoint of NJ.
Prove: MN = 2KL
The complete answer is attached in the diagram below.
The complete answer for the missing reasons is attached below in the diagram.
Please check the figure.
Keywords: statement, proof, reason
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Lisa wants to use her calculator to square a two-digit positive integer, but she accidentally enters the tens digit incorrectly. When she squares the number entered, the result is 2340 greater than the result she would have gotten had she correctly entered the tens digit. What is the sum of the two-digit number Lisa entered and the two-digit number she meant to enter?
Answer:
78
Step-by-step explanation:
You want to know the sum of two 2-digit numbers such that one differs from the other by a multiple of 10, and the difference of their squares is 2340.
Difference of squaresThe difference of squares is the product of the sum and difference of the two numbers. Here, the difference must be a multiple of 10, so we want factorizations of 2340 that have a multiple of 10 as a factor. These are ...
2340 = 10(234) = 20(117) = 30(78)
where the first factor (the difference) is less than the second factor (the sum).
The sum of two 2-digit numbers cannot be more than 200, and it must be even if they both have the same units digit. The only viable product from the above list is 30 × 78, where 30 is the difference of the numbers and 78 is their sum.
The sum of the numbers is 78.
__
Additional comment
The two numbers are 54 and 24. The difference of their squares is ...
2916 -576 = 2340
Difference of squares: a² -b² = (a -b)(a +b).
The cost of controlling emissions at a firm goes up rapidly as the amount of emissions reduced goes up. Here is a possible model: C(x, y) = 7,000 + 100x2 + 50y2 where x is the reduction in sulfur emissions, y is the reduction in lead emissions (in pounds of pollutant per day), and C is the daily cost to the firm (in dollars) of this reduction. Government clean-air subsidies amount to $500 per pound of sulfur and $100 per pound of lead removed. How many pounds of pollutant should the firm remove each day in order to minimize net cost (cost minus subsidy)?
Answer:
2.5 pounds pounds of sulfur and 1 pound of lead should be removed each day in order to minimize net cost.
Step-by-step explanation:
Given model of cost:
[tex]C(x, y) = 7,000 + 100x^2 + 50y^2[/tex]
Government clean-air subsidies amount for sulfur = 500 $/pound
Government clean-air subsidies amount for lead = 100 $/pound
Subsidies amount for x pounds of sulfur = x500 $
Subsidies amount for y pounds of lead= y100 $
Model of subsidy amount :
[tex]S(x,y)=500x+100y[/tex]
Net cost(N) = Cost - Subsidy = C(x,y)-S(x,y)
[tex]N=7,000 + 100x^2 + 50y^2-500x-100y[/tex]..[1]
Differentiating above [1] in with respect to dx :
[tex]\frac{dN}{dx}=\frac{7,000 + 100x^2 + 50y^2-500x-100y}{dx}[/tex]
[tex]\frac{dN}{dx}=200x-500[/tex]..[2]
Putting [tex]\frac{dN}{dx}=0[/tex]:
[tex]0=200x-500[/tex]
x = 2.5
Now taking second derivative of [2]:
[tex]\frac{d^N}{dx^2}=200[/tex]
[tex]\frac{d^N}{dx^2}>0[/tex] (minima)
Differentiating above [1] in with respect to dy :
[tex]\frac{dN}{dy}=\frac{7,000 + 100x^2 + 50y^2-500x-100y}{dy}[/tex]
[tex]\frac{dN}{dy}=100y-100[/tex]..[3]
Putting [tex]\frac{dN}{dy}=0[/tex]:
[tex]0=100y-100[/tex]
y = 1
Now taking second derivative of [3]:
[tex]\frac{d^N}{dy^2}=100[/tex]
[tex]\frac{d^N}{dy^2}>0[/tex] (minima)
2.5 pounds pounds of sulfur and 1 pound of lead should be removed each day in order to minimize net cost.
The amounts of sulfur and lead the firm should remove are 2.5 pounds and 1 pound respectively to minimize the net cost.
Given to us
cost of controlling emissions, C(x, y) = 7,000 + 100x² + 50y²
amount to $500 per pound of sulfur
$100 per pound of lead removed
What is the net cost of the firm?We know that the net cost can be written as,
T(x, y) = 7,000 + 100x² + 50y² -500x -100y
where x and y is the amount of sulfur and lead emission reduced respectively.
What is the minimum amount of sulfur that should be removed?
To find the minimum of x differentiate the value of net cost with respect to x,
[tex]\dfrac{dT}{dx} = \dfrac{ 7,000 + 100x^2 + 50y^2 -500x -100y}{dx}[/tex]
[tex]= \dfrac{ 7,000 + 100x^2 + 50y^2 -500x -100y}{dx}\\\\= 0+ 100(2x) + 0 + 500 + 0\\\\ = 200x+500[/tex]
Substitute against 0, to get the minimum value of x,
0 = 200x+500
x = 2.5
Differentiate again,
[tex]\dfrac{d^2T}{dx^2} = \dfrac{d(200x+500)}{dx}[/tex]
[tex]=200+0[/tex]
As the value of differentiation is positive, therefore, the slope of the function will be going towards the positive.
What is the minimum amount of Lead that should be removed?To find the minimum of y differentiate the net cost with respect to y,
[tex]\dfrac{dT}{dy} = \dfrac{ 7,000 + 100x^2 + 50y^2 -500x -100y}{dy}[/tex]
[tex]= 0+0+50(2y)-100\\\\=100y-100\\\\=100(y-1)[/tex]
Substitute against 0 to get the minimum value of y,
0 = 100(y-1)
y = 1
Differentiate again,
[tex]\dfrac{d^2T}{dy^2} = \dfrac{d(100y-100)}{dy}[/tex]
[tex]=100[/tex]
As the value of differentiation is positive, therefore, the slope of the function will be going towards the positive.
Hence, the amount of sulfur and lead the firm should remove is 2.5 pounds and 1 pound respectively to minimize the net cost.
Learn more about Differentiation:
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The cylinder coffee cup has a radius of 1.8 inches and a height of 4 inches. Find the surface area of the coffee cup, not including the handle. Round to the nearest tenth
Answer:
Step-by-step explanation:
The formula for determining the total surface area of a cylinder is expressed as
Total surface area = 2πr² + 2πrh
Where
r represents the radius of the cylinder.
h represents the height of the cylinder.
π is a constant whose value is 3.14
Assuming that the cylindrical cup is open at the top, the formula becomes
Area = πr² + 2πrh
From the information given,
Radius = 1.8 inches
Height = 4 inches
Therefore, the surface area of the coffee cup is
(3.14 × 1.8²) + (2 × 3.14 × 1.8 × 4)
= 10.1736 + 45.216
= 55.4 inches² to the nearest tenth.
Answer:
55.4 :))
Step-by-step explanation:
60 POINTS AND RAINLIEST!
Koji is installing a rectangular window in an office building. The window is 823 feet wide and 534 feet high.
The formula for the area of a rectangle is A=bh.
What is the area of the window?
Enter your answer as a mixed number in simplest form in the box.
$$
a
ft2
?
? _
?
The area would be 8 2/3* 5 3/4 which according to my calculator is 49 5/6
So the area is 49 5/6 ft^2.
Hope this helped!
The apparent brightness of a star if it were viewed from a distance of 10 parsecs (32.6 light- years) is called ________.
Answer:
Absolute magnitude
Step-by-step explanation: Astronomy deals with the study of stars and other heavenly bodies. Astronomers use apparent magnitude to define how bright a star appears and shines from the earth.
A silicon (Φ = 7.77 × 10-19 J) surface is irradiated with UV radiation with a wavelength of 235 nm. Assume an electron was a mass of 9.11 x 10-31 kg. What is the kinetic energy of the emitted electrons?
Since a silicon (Φ = 7.77 × 10-19 J) surface is irradiated with UV radiation with a wavelength of 235 nm. Assume an electron was a mass of 9.11 x 10-31 kg, the kinetic energy of the emitted electrons is 6.9 × 10⁻²⁰ J
What is kinetic energy of emitted electron in photoelectroic effect?The kinetic energy of emitted electron in photoelectric effect is given by
K = hc/λ - Φ where
h = Planck's constant = 6.63 × 10⁻³⁴ Jsc = speed of light = 3 × 10⁸ m/s λ = wavelength of light andΦ = work function of metalSince a silicon (Φ = 7.77 × 10-19 J) surface is irradiated with UV radiation with a wavelength of 235 nm. Assume an electron was a mass of 9.11 x 10-31 kg. To determine the kinetic energy of the emitted electrons, we proceed as follows
Since the kinetic energy of the emitted electrons is
K = hc/λ - Φ
Given that
λ = 235 nm = 235 × 10⁻⁹ m andΦ = 7.77 × 10⁻¹⁹ JSo, substituting the values of the variables into the equation, we have that
K = hc/λ - Φ
K = (6.63 × 10⁻³⁴ Js × 3 × 10⁸ m/s)/235 × 10⁻⁹ m - 7.77 × 10⁻¹⁹ J
K = (19.89 × 10⁻²⁶ Jm)/235 × 10⁻⁹ m - 7.77 × 10⁻¹⁹ J
K = 0.0846 × 10⁻¹⁷ J - 7.77 × 10⁻¹⁹ J
K = 8.46 × 10⁻¹⁹ J - 7.77 × 10⁻¹⁹ J
K = 0.69 × 10⁻¹⁹ J
K = 6.9 × 10⁻²⁰ J
So, the kinetic energy is K = 6.9 × 10⁻²⁰ J
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Determine whether the data described below are qualitative or quantitative and explain why. The regions of residence of participants in a clinical trial of a new cancer treatment.Choose the correct answer below.A. The data are qualitative because they consist of counts or measurements. B. The data are qualitative because they don't measure or count anything.C. The data are quantitative because they consist of counts or measurements.D. The data are quantitative because they don't measure or count anything.
Answer:
option B
Step-by-step explanation:
The described data is qualitative because regions of a residence has categories and it don't measure or count anything. The regions of residence can be divided into different categories and numerical quantities can be assigned to them but still they will be qualitative variables because numerical quantities will be working as identifiers and they don't measure or count anything.