1)Make x the subject of the formula; x+r=√[(2x+r)^2/4].
2)Make M the subject of the formula; X=-MN/√(m^2+n)
3)Make R the subject of the formula; A=π(R^2-r^2)where A is the area between two concentric circles of radii R and r.
4)Given that L^(1/2)=Pt/p-t and that P=L^2/x,write t in terms of x and l.
5) Make x the subject of the formula; a^2/b=a√(x^2-m/m)
6)In ajua Kali factory the number of pans produced in the first month is 250.The number of pans produced per month increases on the average by 30.Find the expected number of pans produced in the first 12 months
7) Find the sum of the first 30 terms of the series:16+19+22+......
8)A man deposits his money in a savings bank on a monthly basis.Each deposit exceeds the previous one by shs.750.If he started by depositing sh.1500,how much will he have deposited in 12 months.
9)The average of the first and the fourth term of a GP is 140.Given that the first term is 64,find the common ratio.
10)A machine started production of matchboxes at the rate of 12000 per hour.The rate of production decreases by 40% every hour.Calculate the total number of matchboxes in the first two hours.
1) To make x the subject of the formula, we need to isolate x on one side of the equation.
Start by squaring both sides of the equation: (x + r)^2 = (2x + r)^2/4.
Expand both sides: x^2 + 2xr + r^2 = (4x^2 + 4xr + r^2)/4.
Multiply through by 4 to get rid of the fraction: 4x^2 + 8xr + 4r^2 = 4x^2 + 4xr + r^2.
Simplify by canceling out the terms: 8xr + 4r^2 = 4xr + r^2.
Bring like terms to one side: 4xr - 4xr + 4r^2 - r^2 = 0.
Combine like terms: 3r^2 = 0.
Divide both sides by 3: r^2 = 0.
Take the square root of both sides: r = 0.
Finally, substitute the value of r back into the original equation to solve for x: x + 0 = √[(2x + 0)^2/4].
Simplify: x = √[(2x)^2/4].
Simplify further: x = √(4x^2/4).
Simplify again: x = √x^2.
Since x cannot be negative, we take the positive square root: x = x.
Therefore, the solution is x = x.
2) To make M the subject of the formula, we need to isolate M on one side of the equation.
Start by multiplying both sides of the equation by √(m^2 + n): √(m^2 + n)(X) = -MN.
Divide both sides by -N to isolate M: M = -X√(m^2 + n)/N.
Therefore, the solution is M = -X√(m^2 + n)/N.
3) To make R the subject of the formula, we need to isolate R on one side of the equation.
Start by expanding the formula: A = πR^2 - πr^2.
Move πr^2 to the other side: A + πr^2 = πR^2.
Divide both sides by π: (A + πr^2) / π = R^2.
Take the square root of both sides: √[(A + πr^2) / π] = R.
Therefore, the solution is R = √[(A + πr^2) / π].
4) Given that L^(1/2) = Pt / (p - t) and P = L^2 / x, we need to solve for t in terms of x and L.
Start by isolating Pt on one side of the equation: L^(1/2)*(p - t) = Pt.
Expand the equation: pL^(1/2) - Lt^(1/2) = Pt.
Move the Lt^(1/2) term to the other side: pL^(1/2) = Lt^(1/2) + Pt.
Factor out t^(1/2) on the right side: pL^(1/2) = t^(1/2)(L + P).
Divide both sides by t^(1/2): pL^(1/2) / t^(1/2) = L + P.
Raise both sides to the power of 2: (pL^(1/2) / t^(1/2))^2 = (L + P)^2.
Simplify: p^2L / t = L^2 + 2LP + P^2.
Move the L^2 term to the other side: p^2L / t - L^2 = 2LP + P^2.
Multiply through by t to get rid of the fraction: p^2L - Lt = 2L^2P + P^2t.
Move the P^2t term to the other side: p^2L - Lt - P^2t = 2L^2P.
Factor out t: t(p^2L - L - P^2) = 2L^2P.
Divide both sides by (p^2L - L - P^2): t = 2L^2P / (p^2L - L - P^2).
Therefore, the solution is t = 2L^2P / (p^2L - L - P^2).
5) To make x the subject of the formula, we need to isolate x on one side of the equation.
Start by squaring both sides of the equation: (a^2/b)^2 = a^2(x^2 - m/m).
Expand both sides: a^4/b^2 = a^2(x^2 - m).
Divide both sides by a^2: a^2/b^2 = x^2 - m.
Add m to both sides to isolate x: a^2/b^2 + m = x^2.
Take the square root of both sides: √(a^2/b^2 + m) = x.
Therefore, the solution is x = √(a^2/b^2 + m).
6) The expected number of pans produced in the first 12 months can be found by adding up the number of pans produced in each month.
In the first month, 250 pans are produced.
In each subsequent month, the number of pans produced increases by 30.
To find the expected number of pans produced in the first 12 months, we can use the formula for the sum of an arithmetic sequence: Sn = (n/2)(2a + (n-1)d), where Sn is the sum of the first n terms, a is the first term, and d is the common difference.
In this case, a = 250, n = 12, and d = 30.
Plugging in the values: Sn = (12/2)(2(250) + (12-1)(30)) = 6(500 + 330) = 6(830) = 4980.
Therefore, the expected number of pans produced in the first 12 months is 4980.
7) The series can be written as an arithmetic sequence with a common difference of 3: 16, 19, 22, ...
To find the sum of the first 30 terms, we can use the formula for the sum of an arithmetic sequence: Sn = (n/2)(2a + (n-1)d), where Sn is the sum of the first n terms, a is the first term, and d is the common difference.
In this case, a = 16, n = 30, and d = 3.
Plugging in the values: Sn = (30/2)(2(16) + (30-1)(3)) = 15(32 + 29(3)) = 15(32 + 87) = 15(119) = 1785.
Therefore, the sum of the first 30 terms of the series is 1785.
8) Each deposit exceeds the previous one by 750 shillings.
The man started by depositing 1500 shillings.
To find the total amount deposited in 12 months, we can use the formula for the sum of an arithmetic sequence: Sn = (n/2)(2a + (n-1)d), where Sn is the sum of the first n terms, a is the first term, and d is the common difference.
In this case, a = 1500, n = 12, and d = 750.
Plugging in the values: Sn = (12/2)(2(1500) + (12-1)(750)) = 6(3000 + 11(750)) = 6(3000 + 8250) = 6(11250) = 67500.
Therefore, the man will have deposited 67500 shillings in 12 months.
9) The average of the first and fourth term of a geometric progression (GP) is 140.
The first term is given as 64.
To find the common ratio, we can use the formula for finding a term in a GP: aₙ = a₁ * r^(n-1), where aₙ is the nth term, a₁ is the first term, r is the common ratio, and n is the position of the term.
In this case, we are given a₁ = 64 and the average of the first and fourth term is 140.
Setting up the equation: (64 + 140) / 2 = a₁ * r^(4-1).
Simplifying: 102 = 64 * r^3.
Divide both sides by 64: 102 / 64 = r^3.
Simplifying: 1.59 ≈ r^3.
Take the cube root of both sides: ∛(1.59) ≈ r.
Therefore, the common ratio is approximately 1.26.
10) The machine started production at a rate of 12000 matchboxes per hour.
The rate of production decreases by 40% every hour.
To find the total number of matchboxes in the first two hours, we need to add the number of matchboxes produced in the first hour and the number of matchboxes produced in the second hour.
In the first hour, 12000 matchboxes are produced.
In the second hour, the rate of production decreases by 40%, so the machine produces 60% (100% - 40%) of the previous hour's production.
To find the number of matchboxes produced in the second hour, multiply the previous hour's production by 60%: 12000 * 0.6 = 7200 matchboxes.
Adding the matchboxes from both hours: 12000 + 7200 = 19200 matchboxes.
Therefore, the total number of matchboxes in the first two hours is 19200.