H varies as V and inversely as the square of r.Find the percentage change in H if V is increased by 20%and at the same time r is increased by 50%
To find the percentage change in H, we need to determine how the change in V and r affects H.
First, let's establish the equation relating H, V, and r:
H = k * (V / r^2)
where k is the constant of variation.
Now, let's consider the given changes:
1. V is increased by 20%: V_new = V + 0.2V = 1.2V
2. r is increased by 50%: r_new = r + 0.5r = 1.5r
Substitute the new values into the equation:
H_new = k * (V_new / r_new^2)
= k * (1.2V / (1.5r)^2)
= k * (1.2V / 2.25r^2)
= (1.2/2.25) * k * (V / r^2)
= 0.533333... * H
The new value of H is 0.533333... times the original value of H. Therefore, there is a 46.67% decrease in H if V is increased by 20% and r is increased by 50%.
To find the percentage change in H, we can use the concept of direct and inverse variation.
Let's first examine the given information:
H varies as V and inversely as the square of r. This implies that the mathematical equation relating these variables is:
H ∝ V / r^2
Now, let's calculate the individual percentage changes in V and r and see how H is affected.
1. V is increased by 20%:
To calculate the new value of V after an increase of 20%, we can simply multiply the original value of V by 1.2.
New V = V * (1 + 0.2) = 1.2V
Therefore, V has increased by 20% or 0.2V.
2. r is increased by 50%:
To calculate the new value of r after an increase of 50%, we can multiply the original value of r by 1.5.
New r = r * (1 + 0.5) = 1.5r
Therefore, r has increased by 50% or 0.5r.
Now, let's substitute the new values of V and r into the equation to find the new value of H.
New H ∝ (1.2V) / (1.5r)^2 = (1.2V) / (2.25r^2) = (1.2/2.25) * (V/r^2)
Simplifying further, we have:
New H = (0.5333) * (V / r^2)
To find the percentage change in H, we need to compare the new value of H to the original value of H:
Percentage change in H = [(New H - Original H) / Original H] * 100
However, since we're only interested in the sign and not the actual percentage, we can compare the variables in the denominator:
[(0.5333) * (V / r^2) - H] / H
So, the percentage change in H when V is increased by 20% and r is increased by 50% is given by:
[(0.5333) * (V / r^2) - H] / H * 100
Note: The specific numerical value of the percentage change will depend on the initial values of V, r, and H.
To find the percentage change in H, we need to determine the relationship between H, V, and r. The statement "H varies as V and inversely as the square of r" can be written as:
H = k * (V / r^2)
where k is a constant.
Now, let's find the percentage change in H if V is increased by 20% and r is increased by 50%.
First, let's consider the percentage change in V. The increase in V can be represented as:
New V = V + (20% of V) = V + (0.2V) = 1.2V
Next, let's consider the percentage change in r. The increase in r can be represented as:
New r = r + (50% of r) = r + (0.5r) = 1.5r
Now, let's substitute the new values into the equation for H:
New H = k * (1.2V / (1.5r)^2)
To find the percentage change in H, we can calculate:
Percentage change in H = ((New H - H) / H) * 100%
Let's rearrange the equation for New H and substitute it into the formula for percentage change:
Percentage change in H = ((k * (1.2V / (1.5r)^2) - k * (V / r^2)) / k * (V / r^2)) * 100%
Simplifying the equation:
Percentage change in H = ((1.2V / (1.5r)^2) - (V / r^2)) / (V / r^2) * 100%
Now, let's substitute the relationship between V and r:
Percentage change in H = ((1.2V / (1.5r)^2) - (V / r^2)) / (V / r^2) * 100%
Percentage change in H = (((1.2V / (1.5r)^2) - (V / r^2)) * (r^2 / V)) * 100%
Percentage change in H = (1.2 - (1 / 1.5^2)) * 100%
Percentage change in H = (1.2 - (1 / 2.25)) * 100%
Percentage change in H = (1.2 - 0.4444) * 100%
Percentage change in H = 0.7556 * 100%
Percentage change in H ≈ 75.56%
Therefore, the percentage change in H is approximately 75.56% when V is increased by 20% and r is increased by 50%.