Probability of picking an even or prime card at random is approximately 88.89%.
To find the probability of picking an even or prime card at random, we first need to identify the possible cards that meet these conditions.
In a standard deck of cards, there are 52 cards, but we only consider the numerical values, which are 2-10 for each of the four suits (hearts, clubs, spades, and diamonds).
Even numbers: 2, 4, 6, 8, 10
Prime numbers: 2, 3, 5, 7
Combining the even and prime numbers, we have: 2, 3, 4, 5, 6, 7, 8, 10. Note that 2 appears only once in the combined list.
Now we find the probability by dividing the number of favorable outcomes (even or prime numbers) by the total number of possible outcomes (cards numbered 2-10).
P(even or prime) = (Number of even or prime cards) / (Total number of cards from 2-10)
P(even or prime) = 8 / 9
To express the probability as a percentage, multiply by 100:
P(even or prime) = (8 / 9) × 100 ≈ 88.89%
So, the probability of picking an even or prime card at random is approximately 88.89%.
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7. kayla uses her credit card to purchase a new television for $487.89. she can pay off up to $175 per month. the card has an annual rate of 23.5% compounded monthly. how
much will she pay in interest? (2 points)
o $22.78
$156.76
o $8.95
$18.87
save & exit
submit all answers
o
7:42
hp
o
الا : 2
96
5
4
8
backspace
If Kayla can pay off up to $175 monthly for the purchase of a new television for $487.89, the interest she will pay at 23.5% compounded monthly is D. $18.87.
How the compound interest is computed?The compound interest payable on the credit card for the purchase of a new television can be computed using an online finance calculator as follows:
N (# of periods) = 3 months
I/Y (Interest per year) = 23.5%
PV (Present Value) = $487.89
PMT (Periodic Payment) = $-175
Results:
FV = $18.23
Sum of all periodic payments = $525.00
Total Interest = $18.87
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The radius of a circle measures 7 inches. A central angle of the circle measuring 4π15 radians cuts off a sector.
What is the area of the sector?
Enter your answer, as a simplified fraction
The area of sector is 14π/3 square inches for a circle having a radius of 7 inches and measures an angle of 4π/15 radians.
Radius of circle = 7 inches
Angle of circle = 4π/15 radians
The area of a sector of a circle can be calculated by using the formula:
A = (θ/2) × [tex]r^2[/tex]
A = The area of the sector
θ = central angle in radians
r = radius of the circle.
Substituting the given values in the formula:
Area = (θ/2) × [tex]r^2[/tex]
Area = (4π/15 × 1/2) × [tex]7^2[/tex]
Area= (2π/15) × 49
Area = 14π/3
Therefore, we can conclude that the area of the sector is 14π/3 square inches.
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The function f (x) = 15(0.85)^x
models the height, in feet, of a bouncing ball after x seconds.
What is the initial height of the bouncing ball?
What is the percent rate of change?
What is the height of the bouncing ball after 5 seconds? Express your answers as a decimal rounded to the nearest hundredth.
a. The initial height of the bouncing ball is 15 feet.
b. The percent rate of change is 85%.
c. The height of the bouncing ball after 5 seconds is approximately 6.79 feet (rounded to the nearest hundredth).
What is Function ?In mathematics, a function is a rule that assigns each element in a set (the domain) to a unique element in another set (the range). The domain and range can be any sets, but they are typically sets of real numbers.
The function f(x) = 15 (0.85)ˣ models the height, in feet, of a bouncing ball after x seconds.
a. The initial height of the bouncing ball is given by f(0). Plugging in x = 0, we get:
f(0) = 15*1
f(0) = 15(1)
f(0) = 15
Therefore, the initial height of the bouncing ball is 15 feet.
b. The percent rate of change is given by the coefficient of the base, which is 0.85 in this case. To convert this decimal to a percentage, we can multiply by 100:
0.85 × 100 = 85
Therefore, the percent rate of change is 85%.
c. The height of the bouncing ball after 5 seconds is given by f(5). Plugging in x = 5, we get:
f(5) = 15
f(5) ≈ 6.79
Therefore, the height of the bouncing ball after 5 seconds is approximately 6.79 feet (rounded to the nearest hundredth).
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Sam needs 2/5 pound of turkey to make one sandwich he is going to make 7 sandwiches how many pounds of turkey does he need
If Sam needs 2/5 pound turkey to make one sandwich, then to make 7 sandwiches, he will need:
(2/5) x 7 = (2 x 7)/5 = 14/5 = 2.8 pounds of turkey
Therefore, Sam needs 2.8 pounds of turkey to make 7 sandwiches.
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Rachel and nicole are training to run a half marathon. rachel begins by running 30 minutes on the
tirst day of training. each day she increases the time she runs by 3 minutes. nicole's training follows the
function f(x) = 5x + 30, where x is the number of days since the training began, and f(x) is the time in
minutes she runs each day. what is the rate of change in minutes per day for the training program that
has the least rate of change?
rachel:
starting minutes:
increase in rate:
equation:
nicole:
starting minutes:
increase in rate:
equation:
The training program with the least rate of change is Rachel's, with an increase of 3 minutes per day.
Rachel:
Starting minutes: 30
Increase in rate: 3 minutes per day
Equation: f(x) = 3x + 30
Nicole:
Starting minutes: 30 (since f(0) = 5(0) + 30 = 30)
Increase in rate: 5 minutes per day
Equation: f(x) = 5x + 30
To find the training program with the least rate of change, we need to find the derivative of each equation and set it equal to zero:
f'(x) = 3 for Rachel's equation
f'(x) = 5 for Nicole's equation
Since 3 is less than 5, Rachel's training program has the least rate of change. Therefore, the rate of change in minutes per day for Rachel's training program that has the least rate of change is 3 minutes per day.
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In two or more complete sentences, describe the steps a consumer can take to become more knowledgeable.
uploa
There are several steps a consumer market can take to become more knowledgeable like research and asking questions.
Research The first step is to probe the product or service that you're interested in. This involves looking at reviews, product descriptions, and comparing prices. You can also look for information from dependable sources similar as consumer reports or government websites. Ask questions If you have any dubieties or enterprises, don't vacillate to ask the dealer or service provider.
Ask them about their experience and qualifications, and make sure to clarify any terms or conditions that are unclear. Get a alternate opinion If you're doubtful about a product or service, seek the advice of someone you trust or who has moxie in that area. They can help you make an informed decision grounded on their knowledge and experience.
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the life of light bulbs is distributed normally. the standard deviation of the lifetime is 20 hours and the mean lifetime of a bulb is 580 hours. find the probability of a bulb lasting for at most 624 hours. round your answer to four decimal places.
The probability of a bulb lasting for at most 624 hours is 0.9861, rounded to four decimal places.
The standard deviation of the lifetime is a measure of how spread out the lifetimes are. In other words, it tells us how much the lifetimes of bulbs vary from the mean. In this case, the standard deviation of the lifetime is 20 hours.
Now, let's get to the question at hand. We want to find the probability of a bulb lasting for at most 624 hours. To do this, we need to use the properties of the normal distribution.
First, we need to calculate the z-score, which tells us how many standard deviations a value is from the mean. We can use the formula z = (x - mu) / sigma, where x is the value we are interested in, mu is the mean, and sigma is the standard deviation. In this case, x = 624, mu = 580, and sigma = 20.
Plugging these values into the formula, we get z = (624 - 580) / 20 = 2.2.
Next, we need to find the probability of a bulb lasting for at most 624 hours, which is the same as finding the area under the normal curve to the left of z = 2.2. We can use a standard normal distribution table or a calculator to find this probability.
Using a calculator, we can use the normal cdf function with the values -9999 (a very large negative number) and 2.2 to find the probability. This gives us a probability of 0.9861.
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A watch designer claims that men have wrist breadths with a mean equal to 9 cm. A simple random sample of wrist breadths
of 72 men has a mean of 8.91 cm. The population standard deviation is 0.36 cm.
Assume a confidence level of a = 0.01. Find the value of the test statistic z using
formula below
Z=
X-H
σ
O2.12
O-1.27
O 0.06
O-2.12
The value of the test statistic z is approximately -2.12. The Option D is correct.
What is the value of the test statistic z?To test the hypothesis that the mean wrist breadth of men is equal to 9 cm, we will use a one-sample z-test.
The null hypothesis is: H0: µ = 9 cm
The alternative hypothesis is: Ha: µ ≠ 9 cm
We are given a sample of n = 72 men with a sample mean of x = 8.91 cm and a population standard deviation = 0.36 cm.
The test statistic for a one-sample z-test is given by: z = (x - µ) / (o / sqrt(n))
Substituting the given values, we get:
= (8.91 - 9) / (0.36 / sqrt(72))
z = -2.119
At a significance level of a = 0.01, the critical values for a two-tailed test are ±2.576.
Since our test statistic (-2.119) falls outside of this range, we reject the null hypothesis and conclude that there is sufficient evidence to suggest that the mean wrist breadth of men is not equal to 9 cm.
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A store has 25 VCRs in stock, but 2 of these are defective. What is the probability
that the second person to buy a VCR gets a defective one and the first
customer's VCR was not defective? Round your answer to the nearest
thousandth. *
. 083
. 0736
. 077
. 08
A store has 25 VCRs in stock, but 2 of these are defective he answer is the probability that the second person to buy a VCR gets a defective one and the first customer's VCR was not defective is .077.
The probability that the first customer's VCR is not defective is 23/25, as there are 23 working VCRs out of the total 25.
Since one VCR has already been sold, there are 24 VCRs left and 1 defective VCR. Thus, the probability that the second customer gets a defective VCR is 1/24.
To find the probability that both events occur, we multiply the individual probabilities:
P = (23/25) x (1/24)
P = 0.077 or 0.0778 when rounded to the nearest thousandth.
Therefore, A store has 25 VCRs in stock, but 2 of these are defective he answer is the probability that the second person to buy a VCR gets a defective one and the first customer's VCR was not defective is .077.
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The original selling price of a jacket was
s
s dollars. The selling price was then changed on two occasions by the store owner. Its price is now represented by
0. 85
(
1. 4
s
)
0. 85(1. 4s). Which expression could explain what happened to the price of the jacket?
The expression 0.85(1.4s) represents the final selling price of the jacket after two price changes: an initial 15% decrease in price, followed by another 15% decrease in price.
Find out which expression could explain the happened price of the jacket?The expression 0.85(1.4s) represents the final selling price of the jacket after two price changes. We can break it down into its constituent parts to understand what happened to the price.
Factor 1.4s represents the original selling price of the jacket. This means that the store owner started with a price of 1.4s dollars.
The factor 0.85 represents the first price change. When the store owner lowered the price by 15%, the new price became 0.85 times the original price.
The second factor of 0.85 represents the second price change. After the first price change, the new price was 0.85(1.4s) dollars. When the store owner lowered the price again by 15%, the final selling price became 0.85 times the price after the first price change.
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How much interest has accrued after one month at a rate of 15. 5%? Use the formula I=Prt. *
A $19. 24
B $18. 71
C $18. 81
The interest accrued after one month at a rate of 15.5% on a principal amount of $1,000 is $12.92.
To use the formula I=Prt to calculate the interest accrued after one month at a rate of 15.5%, we need to know the principal amount (P) and the time period (t) in years.
Assuming that the principal amount is $1,000, and the time period is one month, which is equivalent to 1/12 of a year, we can calculate the interest as follows:
[tex]I = Prt[/tex]
[tex]I[/tex] [tex]= 1000 x 0.155 x (1/12)[/tex]
[tex]I = $12.92[/tex]
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1pt A clothing company needs to determine how much fabric to use for a sleeve on a shirt. It uses the following model arm as a way to test the fit. The sleeve needs to cover
the 20 centimeters from the shoulder to the elbow.
Upper Arm
What solid best represents the model for the sleeve? What is the minimum surface area of fabric needed for the sleeve?
The minimum surface area of fabric needed for the sleeve is approximately 628.32 cm².
How much fabric for sleeve?Based on the given model arm, a right circular cylinder would best represent the model for the sleeve.
To calculate the minimum surface area of fabric needed for the sleeve, we need to find the lateral surface area of the right circular cylinder.
The lateral surface area of a right circular cylinder is given by the formula:
Lateral surface area = 2πrh
where r is the radius of the cylinder, h is the height of the cylinder.
In this case, the height of the cylinder needs to be 20 cm (to cover the distance from the shoulder to the elbow), and the radius can vary depending on the desired fit. Let's assume a radius of 5 cm for the purposes of this calculation.
Plugging in the values, we get:
Lateral surface area = 2π(5 cm)(20 cm)
= 628.32 cm²
Therefore, the minimum surface area of fabric needed for the sleeve is approximately 628.32 cm².
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1) Amy deposits $2,300 in an account that pays 8.5% interest. How much money will Amy have after 4 years?
2)Andres deposits $10,000 in an account that pays 8% interest. How much money will Andres have after 4 years?
a
$ 13,604.89
b
$ 604.90
c
$ 20,004.98
3) Kara deposits $500 in an account that pays 5% interest. How much money will Kara have after 2 years?
a
$ 1,009.34
b
$ 13.97
c
$ 551.25
1) If Amy deposits $2,300 in an account that pays 8.5% interest, after 4 years, the future value will be $3,187.48.
2) If Andres deposits $10,000 in an account that pays 8% interest, after 4 years, the future value will be A. $13,604.89.
3) If Kara deposits $500 in an account that pays 5% interest, after 2 years, the future value will be C. $551.25.
How the future values are determined:The future values represent the present investment compounded at an interest rate.
The future values can be determined using an online finance calculator as follows:
1) N (# of periods) = 4 years
I/Y (Interest per year) = =8.5%
PV (Present Value) = $2,300
PMT (Periodic Payment) = $0
Results:
Future Value (FV) = $3,187.48
Total Interest = $887.48
2) N (# of periods) = 4 years
I/Y (Interest per year) = =8%
PV (Present Value) = $10,000
PMT (Periodic Payment) = $0
Results:
Future Value (FV) = $13,604.89
Total Interest = $3,604.89
3) N (# of periods) = 2 years
I/Y (Interest per year) = 5%
PV (Present Value) = $500
PMT (Periodic Payment) = $0
Results:
Future Value (FV) = $551.25
Total Interest = $51.25
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11. April shoots an arrow upward at a speed
of 80 feet per second from a platform 25
feet high. The pathway of the arrow can
be represented by the equation h =-
16t2 + 80t + 25, where h is the height
and t is the time in seconds. What is the
maximum height of the arrow? [3]
The maximum height of the arrow is 105 feet. To find the maximum height of the arrow, we need to determine the vertex of the quadratic function h = -16[tex]t^{2}[/tex] + 80t + 25.
The vertex is the highest point on the graph of the function, which represents the maximum height of the arrow.
To find the t-value at the vertex, we use the formula t = -b/2a, where a = -16 and b = 80. Plugging these values into the formula gives us t = -80/(2(-16)) = 2.5 seconds.
To find the maximum height, we plug t = 2.5 into the equation to get h = -16[tex](2.5)^{2}[/tex] + 80(2.5) + 25 = 105 feet. Therefore, the maximum height of the arrow is 105 feet.
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what is the product of -5 and -10 sign and result
Answer:
50, positive
Step-by-step explanation:
(-5) * (-10) = 50
When a negative multiplies by another negative, the answer is positive
So, the answer is 50 and the sign is positive
pls some help with this question!
Which of these variables is your dependent variable?
How many jumps I can do
Which one is the independent variable?
How long I am jumping (2 minutes)
Write a sentence that describes the relationship between the dependent variable and the independent variable. (Hint: Ratio language can help. )
In this scenario, the dependent variable is "how many jumps I can do," while the independent variable is "how long I am jumping (2 minutes)."
The relationship between these variables can be described as follows: The number of jumps completed depends on the duration of time spent jumping, with a specific focus on a 2-minute interval.
When we say the dependent variable is "how many jumps I can do," it means that the number of jumps completed is determined by or depends on the independent variable, which is the duration of time spent jumping.
This suggests that as the duration of time increases or decreases, it will likely have an impact on the number of jumps performed.
In this particular case, you have specified a 2-minute interval as the focus. It suggests that you are examining the relationship between the number of jumps completed and the specific duration of 2 minutes.
This implies that you are interested in understanding how the number of jumps varies within this fixed time frame.
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Write an exponential function to model the following situation.
a population of 140,000 grows 5% per year for 15 years.
how much will the popluation be after 15 years?
write an exponential function in terms of x.
An exponential function in terms of x is [tex]P(x) = 140,000(1.05)^x[/tex]
The population would be 291049.95 after 15 years.
How to determine the population after a number of year?In Mathematics, a population that increases at a specific period of time represent an exponential growth. This ultimately implies that, a mathematical model for any population that increases by r percent per unit of time is an exponential function of this form:
[tex]P(x) = I(1 + r)^x[/tex]
Where:
P(t ) represent the population.x represent the time or number of years.I represent the initial number of persons.r represent the exponential growth rate.By substituting given parameters, we have the following:
[tex]P(x) = 140,000(1 + 0.05)^x\\\\P(x) = 140,000(1.05)^x[/tex]
After 15 years, we have:
[tex]P(15) = 140,000(1.05)^{15}[/tex]
P(15) = 291049.95 units.
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Mrs. Rambo got a YMCA membership for her family. The pass has a onetime fee of $30 and then $5 for every visit to the YMCA. Her bill the first month was $150. How many times did her family visit the YMCA?
They visited ------------- times last month. Way to go Rambo family!
Answer:
Step-by-step explanation: Solution:
Total cost: 150
150-30=120
120 divided by 5 = 24
They visited 24 times in a month.
Question 2(Multiple Choice Worth 4 points) (05.03 MC) Solve the system of equations using elimination. 2x + 3y = -8 3x+y=2 O(-4,0) (2,-4) (5.-6) (8-8)
Answer:
(2,-4)
Step-by-step explanation:
In order to solve by elimination, coefficients of one of the variables must be the same in both equations so that the variable will cancel out when one equation is subtracted from the other.
2x+3y=−8,3x+y=2
To make 2x and 3x equal, multiply all terms on each side of the first equation by 3 and all terms on each side of the second by 2. Then simplify
6x+9y=−24,6x+2y=4
Add 6x to −6x. Terms 6x and −6x cancel out, leaving an equation with only one variable that can be solved, add 9y to −2y, add −24 to −4, and divide both sides by 7.
y=−4
Substitute −4 for y in 3x+y=2. Because the resulting equation contains only one variable, you can solve for x directly. Add 4 to both sides of the equation and divide both sides by 3.
x=2
HELP ASAP!!!!!!!!!!!
Answer:
25%
Step-by-step explanation:
The total number of 7th grade students = 9 + 11 + 11 + 13 = 44
Out of the 44 students 11 play bass
Probability that a seventh grader chosen at random will play the base is:
11/44 = 1/4 = 0.25
As a percentage, this would be 0.25 x 100 = 25%
KLM has vertices K 4,-5 L 2,2 and M 7,3 which translation move the triangle so that point K lies on the Y axis
To move triangle KLM so that point K lies on the Y-axis, you need to apply a translation that shifts the entire triangle horizontally. By translating triangle KLM using the vector (-4, 0), point K now lies on the Y-axis.
To move the triangle so that point K lies on the Y axis, we need to perform a translation. First, we need to determine how far point K is from the Y axis. We can do this by finding the x-coordinate of point K, which is 4. This means that point K is 4 units away from the Y axis.
Next, we need to determine the direction of the translation. Since we want to move point K onto the Y axis, we need to move the triangle in the negative x direction. Therefore, the translation that will move the triangle so that point K lies on the Y axis is a horizontal translation of -4 units. We can express this translation as follows:
T(-4, 0)
This means that we need to move each point of the triangle 4 units to the left (negative x direction) to achieve the desired position of point K on the Y axis.
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Luis tiene una mochila de ruedas
que mide 3.5 pies de alto cuando se extiende el mango.Al hacer rodar
su mochila, la mano de Luis se
encuentra a 3 pies del suelo. ?Que ángulo forma su mochila con el suelo? Aproxima al grado más cercano
As a result, the angle formed by Luis's backpack and the ground is roughly 53 degrees (rounded to the nearest degree).
what is function ?A function in mathematics is a relationship between a set of potential outputs and a number of potential inputs, with the feature that each input is associated to only one possible output. It is a principle or set of guidelines that allots a different output value towards each input value. Equations, graphs, and tables are frequently used to depict functions in order to explain how the output changes as the input does. They are employed to express relationships between various quantities, such as the length of time it takes for an automobile to drive a certain distance or the height of an object in relation to its weight. The concept of a function is crucial to many departments of science and math, and it is widely applied in areas like engineering,
given
We can use trigonometry to determine the angle that Luis's backpack creates with the ground.
Consider the backpack's handle to represent the hypotenuse of a right triangle, with the vertical leg being Luis's hand's distance from the ground (3 feet) and the horizontal leg being the backpack's height (3.5 feet).
The angle can be determined using the inverse tangent function (tan-1):
53.13 degrees at tan-1(3.5/3)
As a result, the angle formed by Luis's backpack and the ground is roughly 53 degrees (rounded to the nearest degree).
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From the following facts, complete a depreciation schedule by using
the straight-line method:
Cost of Honda Account Hybrid - $40000
Residual Value - $10000
Estimated Life - 6 years
Using the straight-line method, we can find the depreciation expense per year by dividing the depreciable value (cost - residual value) by the estimated life:
Depreciable Value = Cost - Residual Value
Depreciable Value = $40000 - $10000
Depreciable Value = $30000
Annual Depreciation Expense = Depreciable Value / Estimated Life
Annual Depreciation Expense = $30000 / 6
Annual Depreciation Expense = $5000
To create a depreciation schedule, we can subtract the annual depreciation expense from the cost each year until we reach the residual value:
| Year | Cost | Depreciation | Accumulated Depreciation | Book Value |
|------|---------------|----------------- |----------------------------------------|------------|
| 1 | $40000 | $5000 | $5000 | $35000 |
| 2 | $35000 | $5000 | $10000 | $30000 |
| 3 | $30000 | $5000 | $15000 | $25000 |
| 4 | $25000 | $5000 | $20000 | $20000 |
| 5 | $20000 | $5000 | $25000 | $15000 |
| 6 | $15000 | $5000 | $30000 | $10000 |
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CDE is a tangent to the circle below.
Calculate the size of angle θ.
Fully Justify your answer.
Applying the inscribed angle theorem, the measure of the size of angle ∅ = 85 degrees.
How to Apply the Inscribed Angle Theorem?If an inscribed angle in a circle is subtended by an arc, the inscribed angle theorem states that the measure of the intercepted arc would be twice the measure of the inscribed angle.
Therefore, we have:
measure of arc DF = 2(31) = 62 degrees [inscribed angle theorem]
measure of arc BD = 2(54) = 108 degrees.[inscribed angle theorem]
∅ = 1/2(measure of arc BDF) [inscribed angle theorem]
∅ = 1/2(m(DF) + m(BD))
Substitute:
∅ = 1/2(62 + 108)
∅ = 1/2(170)
∅ = 85 degrees.
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Question 3 of 10
What property does the equation show?
32 19
14
56
1 3
+
=
A. The associative property
B. The commutative property
OC. The distributive property
32 19
14 56
D. The identity property of multiplication
5 8
1 3
+
32
14
19
56
4
7
Answer:
C. The distributive property
Based on the experiment if the spinner is spun 150 times how many times would you expect to get an even number?
Answer:
60
Step-by-step explanation:
((sum of frequency of even numbers)/(total number of tries))(150)
A can is to be made to hold a litre of oil. Find the radius of the can that will minimize the cost of the metal to make the can. (1L = 1000 cm)
The problem involves finding the radius of a cylindrical can that will minimize the cost of the metal to make the can, given that the can must hold one liter of oil.
Specifically, we need to find the radius of the can that will minimize the surface area, and hence the cost, of the metal required to make the can.
To solve the problem, we need to first write an expression for the surface area of the can in terms of its radius, and then differentiate this expression with respect to the radius to find the critical point. We then need to check that the critical point corresponds to a minimum value of the surface area, which will give us the optimal radius for the can. Optimization problems like this one are used in many fields, including engineering, economics, and physics, to find the best course of action given certain constraints and objectives.
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circles P and Q are tangent to eachother and to the axis shown. PQ = 26 and AB = 24. Find the coordinates of P and the coordinates of Q.
The coordinates of P and Q are P(5, 5) and Q(7, 7) respectively.
Understanding TangentLet the centres of the circles be:
P (a, r) and
Q (b, s)
where r and s are the radii of the circles.
Since the circles are tangent to the x-axis, we know that r = a and s = b.
Also, since the circles are tangent to each other, we have
a + b = PQ = 26
Let the point of contact of circle P with the x-axis be (p, 0)
Let the point of contact of circle Q with the x-axis be (q, 0).
Then, we know that
p + q = AB = 24
Using Pythagorean theorem, we can write:
(r² - p²) + (r² - (24 - p)²) = (s²- q²) + (s² - (24 - q)²)
Expanding and simplifying, we get:
2r² - 24r + 576 = 2s² - 24s + 576
Substituting r = a and s = b, and using the fact that a + b = 26, we get:
2a² - 24a + 576 = 2b² - 24b + 576
Simplifying further, we get:
a² - 12a + 288 = b² - 12b + 288
(a - b)(a + b - 12) = 0
Since a + b = 26, we have a - b = 0 or a + b - 12 = 0. The first case gives us a = b, which is not possible since the circles are tangent to each other. Therefore, we have a + b = 12.
Using substitution method to solve the simultaneous equations:
a + b = 12
a + b = 26
We get a = 7 and b = 5.
Therefore, the centres of the circles P and Q are (7, 7) and (5, 5) respectively.
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