1)P varies directly as the square of Q and inversely as the square root of R.If Q is reduced by 12% while R is increased by 21%,find the percentage change in P.
2)Two variables p and q are such that p is directly proportional to x and q is inversely proportional to x.When x=2,their sum is 8.When x=3,their sum is 7.Find the constants of the proportionality.
3)A quantity P varies jointly as the square root of Q and as the inverse of the square of R.Determine the percentage change in P when Q is increased by 44% and R is decreased by 28%.
4)The mass of wire m grams(g) is partly constant and partly varies as the square of its thickness t mm.When t=2mm, m=40g and when t=4 mm, m=100g.Determine the value of m when t=7mm.
5)Make n the subject of the formula; m=3^√ax-n/w-n
1) Let's denote the initial values of Q and R as Q₀ and R₀, and the initial value of P as P₀.
We have the following equations:
P = k₁ * Q² / √R
P₀ = k₁ * Q₀² / √R₀, where k₁ is a constant of proportionality
If Q is reduced by 12%, the new value of Q is Q = 0.88Q₀.
If R is increased by 21%, the new value of R is R = 1.21R₀.
We want to find the percentage change in P, which we can calculate using:
Percentage change = (P - P₀) / P₀ * 100
Substituting the new values of Q and R into the equation for P, we get:
P = k₁ * (0.88Q₀)² / √(1.21R₀)
P₀ = k₁ * Q₀² / √R₀
Calculating the new value of P and using the percentage change formula, we have:
Percentage change = (k₁ * (0.88Q₀)² / √(1.21R₀) - k₁ * Q₀² / √R₀) / (k₁ * Q₀² / √R₀) * 100
Percentage change = (0.88² / √1.21 - 1) / 1 * 100
Percentage change = (0.7744 / 1.1 - 1) * 100
Percentage change = (0.704 - 1) * 100
Percentage change = -29.6%
Therefore, the percentage change in P is -29.6%.
2) Let's denote the constants of proportionality as k₁ and k₂.
We have the following equations:
p = k₁ * x
q = k₂ / x
When x = 2, the sum of p and q is 8:
p + q = 8
k₁ * 2 + k₂ / 2 = 8
Similarly, when x = 3, the sum of p and q is 7:
p + q = 7
k₁ * 3 + k₂ / 3 = 7
We can solve these two equations to find the constants k₁ and k₂.
2k₁ + k₂ / 2 = 8 (equation 1)
3k₁ + k₂ / 3 = 7 (equation 2)
Multiplying equation 1 by 2 and equation 2 by 3 to eliminate the fractions:
4k₁ + k₂ = 16 (equation 3)
9k₁ + k₂ = 21 (equation 4)
Subtracting equation 3 from equation 4 to eliminate k₂:
9k₁ + k₂ - (4k₁ + k₂) = 21 - 16
5k₁ = 5
k₁ = 1
Substituting k₁ = 1 into equation 3, we get:
4(1) + k₂ = 16
k₂ = 12
Therefore, the constants of proportionality are k₁ = 1 and k₂ = 12.
3) Let's denote the initial values of Q and R as Q₀ and R₀, and the initial value of P as P₀.
We have the following equations:
P = k₁ * √Q * 1/R²
P₀ = k₁ * √Q₀ * 1/R₀², where k₁ is a constant of proportionality
If Q is increased by 44%, the new value of Q is Q = 1.44Q₀.
If R is decreased by 28%, the new value of R is R = 0.72R₀.
We want to find the percentage change in P, which we can calculate using:
Percentage change = (P - P₀) / P₀ * 100
Substituting the new values of Q and R into the equation for P, we get:
P = k₁ * √(1.44Q₀) * 1/(0.72R₀) = 1.44 * (k₁ * √Q₀ * 1/R₀²)
P₀ = k₁ * √Q₀ * 1/R₀²
Calculating the new value of P and using the percentage change formula, we have:
Percentage change = (1.44 * (k₁ * √Q₀ * 1/R₀²) - k₁ * √Q₀ * 1/R₀²) / (k₁ * √Q₀ * 1/R₀²) * 100
Percentage change = (1.44 - 1) / 1 * 100
Percentage change = 0.44 * 100
Percentage change = 44%
Therefore, the percentage change in P is 44%.
4) Let's denote the value of m when t = 2mm as m₀.
We have the following equations:
m = c₁ + c₂ * t²
m₀ = c₁ + c₂ * (2)² = c₁ + 4c₂ = 40
When t = 4mm,
m = c₁ + c₂ * (4)² = c₁ + 16c₂ = 100
Subtracting the equation for m₀ from the equation for m when t = 4mm, we get:
c₁ + 16c₂ - (c₁ + 4c₂) = 100 - 40
12c₂ = 60
c₂ = 5
Substituting c₂ = 5 into the equation for m₀, we have:
c₁ + 4(5) = 40
c₁ + 20 = 40
c₁ = 20
Therefore, when t = 7mm, the value of m is:
m = c₁ + c₂ * (7)² = 20 + 5 * 49 = 20 + 245 = 265g
So, when t = 7mm, the value of m is 265g.
5) To make n the subject of the formula m = 3^√(ax - n) / (w - n), we can follow these steps:
Multiply both sides of the equation by (w - n):
m * (w - n) = 3^√(ax - n)
Take the logarithm (base 3) of both sides of the equation:
log₃(m * (w - n)) = √(ax - n)
Square both sides of the equation:
[log₃(m * (w - n))]² = ax - n
Rearrange the equation to isolate n:
n = ax - [log₃(m * (w - n))]²
Therefore, n is the subject of the formula n = ax - [log₃(m * (w - n))]².