what is the common ratio of the geometric progression (√ 2-1)+(3-2 √ 2)+...?
Find the third term of the progression
To find the common ratio of the given geometric progression, we need to divide each term by its preceding term:
(3 - 2√2) ÷ (√2 - 1) = (3 - 2√2) × (√2 + 1) ÷ (√2 - 1) × (√2 + 1)
= (3√2 + 3 - 4√2 - 2) ÷ (2 - 1)
= √2 + 1
So, the common ratio of the given geometric progression is √2 + 1.
To find the third term, we can either calculate it directly using the formula for the nth term of a geometric progression or by multiplying the first two terms by the common ratio:
(√2 - 1) × (√2 + 1) × (√2 + 1)
= (√2 - 1) × (2 + 2√2 + 1)
= (√2 - 1) × (3 + 2√2)
= 3√2 - 5.
Therefore, the third term of the given geometric progression is 3√2 - 5.
a number is made up of two digits. the sum of the digits is 11. if the digits are interchanged, the original number is increased by 9. find the number
Let the original number be 𝑎𝑏, where 𝑎 is the tens digit and 𝑏 is the units digit. From the problem statement, we know that:
𝑎 + 𝑏 = 11 (1)
When the digits are interchanged, the new number is 𝑏𝑎, which is 9 more than the original number:
𝑏𝑎 = 𝑎𝑏 + 9 (2)
We can simplify equation (2) by expanding the products:
10𝑏 + 𝑎 = 10𝑎 + 𝑏 + 9
Simplifying and rearranging terms, we get:
9𝑎 − 9𝑏 = 9
Dividing both sides by 9, we obtain:
𝑎 − 𝑏 = 1 (3)
Now we have two equations ((1) and (3)) with two unknowns (𝑎 and 𝑏) that we can solve simultaneously. Adding equations (1) and (3), we get:
2𝑎 = 12
Therefore, 𝑎 = 6. Substituting this value of 𝑎 into equation (1), we get:
6 + 𝑏 = 11
So 𝑏 = 5. Therefore, the original number is 65.
eight wooden poles are to be used for pillars and the lengths of the poles form an arithmetic progression. if the second pole is 2m and the sixth is 5m, give the lengths of the poles in order
Let the common difference between the consecutive lengths of the poles be 𝑑, and let the length of the second pole (the first term) be 𝑎. Then the lengths of the poles can be expressed as:
𝑎, 𝑎 + 𝑑, 𝑎 + 2𝑑, 𝑎 + 3𝑑, 𝑎 + 4𝑑, 𝑎 + 5𝑑, 𝑎 + 6𝑑, 𝑎 + 7𝑑
We know that the length of the second pole is 2m, so 𝑎 + 𝑑 = 2.
Also, the length of the sixth pole is 5m, so 𝑎 + 5𝑑 = 5.
We can use these two equations to solve for 𝑎 and 𝑑. Rearranging the first equation, we get 𝑎 = 2 − 𝑑.
Substituting this into the second equation, we get:
(2 − 𝑑) + 5𝑑 = 5
Expanding and simplifying, we get:
4𝑑 = 3
Therefore, 𝑑 = 0.75.
Substituting this value of 𝑑 into 𝑎 + 𝑑 = 2, we get:
𝑎 + 0.75 = 2
So 𝑎 = 1.25.
Therefore, the lengths of the poles in order are :
1.25m, 2.00m, 2.75m, 3.50m, 4.25m, 5.00m, 5.75m, 6.50m.
in the diagram above PQ//RS//.T^FS=(7m-16), given that x:y =2:7. find the value of m
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question
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a market woman purchased a number of plates for GH$150.00. four of the plates got broken while transporting them to her shop. By selling the remaining plates at a profit of GH$6.00. how many plates did she sell?
Let 𝑥 be the number of plates the market woman purchased.
If 4 of the plates got broken, then she had 𝑥 − 4 plates left to sell.
Let 𝑐 be the cost of each plate. Then we can write the equation:
𝑥𝑐 = 150
Dividing both sides by 𝑥, we get:
𝑐 = 150/𝑥
Selling the remaining plates at a profit of GH$6.00 means that she sold them for:
𝑐 + 6
The total amount of sales was therefore:
(𝑥 − 4)(𝑐 + 6)
We know that the total amount of sales was greater than or equal to the cost, or:
(𝑥 − 4)(𝑐 + 6) ≥ 𝑥𝑐
Expanding both sides, we get:
𝑥𝑐 + 6𝑥 - 4𝑐 - 24 ≥ 𝑥𝑐
Simplifying and rearranging terms, we get:
6𝑥 - 4𝑐 - 24 ≥ 0
Substituting 𝑐 = 150/𝑥, we get:
6𝑥 - 4(150/𝑥) - 24 ≥ 0
Multiplying both sides by 𝑥, we get:
6𝑥^2 - 600 - 24𝑥 ≥ 0
Dividing both sides by 6, we get:
𝑥^2 - 4𝑥 - 100 ≥ 0
We can solve this inequality by factoring:
(𝑥 - 10)(𝑥 + 6) ≥ 0
Thus, 𝑥 ≥ 10 or 𝑥 ≤ −6.
Since the number of plates cannot be negative, we have 𝑥 ≥ 10.
Therefore, the market woman purchased at least 10 plates.
Substituting 𝑥 = 10 into 𝑥𝑐 = 150, we get 𝑐 = 15.
Thus, each plate cost GH$15.00.
Selling the remaining (10 − 4) = 6 plates at a profit of GH$6.00 per plate, the total profit is 6(15 + 6) = GH$126.00.
The total amount of sales was therefore GH$150.00 + GH$126.00 = GH$276.00.
To find the number of plates sold, we can divide this by the selling price per plate:
𝑥 − 4 = 276/(15 + 6) = 12
Therefore, the market woman sold 12 plates.
a box contains identical balls of which 12 are red, 16 white and 8 blue. Three balls are drawn from the box one after the other without replacement. Find the probability that:
i.three are red
ii.the first is blue and the other two are red
iii.two are white and one is blue
Total number of balls in the box = 12 + 16 + 8 = 36
i. Probability that all three balls are red:
P(3 red) = (12/36) × (11/35) × (10/34) = 0.02381 (rounded to 5 decimal places)
ii. Probability that the first ball is blue and the other two are red:
P(blue then 2 red) = (8/36) × (12/35) × (11/34) = 0.01426 (rounded to 5 decimal places)
iii. Probability that two balls are white and one is blue:
P(2 white and 1 blue) = [(16/36) × (15/35) × (8/34)] + [(16/36) × (8/35) × (15/34)] + [(8/36) × (16/35) × (15/34)]
= 0.33095 (rounded to 5 decimal places)
Therefore, the probability that three balls drawn from the box, without replacement, are as specified are:
i. Probability of three red balls = 0.02381
ii. Probability of one blue and two red balls = 0.01426
iii. Probability of two white and one blue balls = 0.33095
from an aeroplane in the air and at a horizontal distance of 1050m,the angles of depression of the top and base of a control tower at an instant are 36 and 41 respectively.
calculate, correct to the nearest metre the;
i.height of the control tower
ii.shortest distance between the aeroplane and the base of the control tower
Let 𝑥 be the height of the control tower above the ground and let 𝑑 be the horizontal distance between the plane and the base of the tower.
Then, we have two right triangles: a larger triangle with the plane at the right angle and a smaller triangle with the tower at the right angle:
- In the larger triangle, the opposite side from the angle of depression of 36 degrees is 𝑥, and the adjacent side is 𝑑 + 1050m.
- In the smaller triangle, the opposite side from the angle of depression of 41 degrees is 𝑥 − 1050 tan(36°), and the adjacent side is 𝑑.
We can use the tangent function to write two equations involving 𝑥 and 𝑑:
𝑥/(𝑑 + 1050) = tan(36°)
(𝑥 − 1050 tan(36°))/𝑑 = tan(41°)
Solving for 𝑥 in the first equation, we get:
𝑥 = (𝑑 + 1050) tan(36°)
Substituting this into the second equation, we get:
(𝑑 + 1050) tan(36°) − 1050 tan(36°) tan(41°) = 𝑑 tan(41°)
Expanding and simplifying, we get:
𝑑 tan(36°) − 1050 tan(36°) tan(41°) = −1050 tan(41°)
Dividing by tan(36°), we get:
𝑑 − 1050 tan(41°) = −1050 tan(41°)/tan(36°)
Simplifying, we get:
𝑑 = (−1050 tan(41°)/tan(36°)) + 1050 tan(41°) = 2143.6335 m (rounded to 4 decimal places)
Therefore, the shortest distance between the plane and the base of the control tower is approximately 2143.6335 m.
Substituting this into the first equation, we get:
𝑥 = (2143.6335 + 1050) tan(36°) = 1303.4007 m (rounded to 4 decimal places)
Therefore, the height of the control tower is approximately 1303.4007 m.