P varies partly as q and partly as the square of R. When P =6,q =12,R=4 and when P=10, q=12, R=5.a: find the formula connecting P, q and R . b : find the value of P when q=15, R=25.c: find the value of R when q=4 and P=2.5 .
p = aq + br^2
Now use your data points to get
12a+16b = 6
12a+25b = 10
I suspect a typo
To find the formula connecting P, q, and R, we can use the given information that P varies partly as q and partly as the square of R. Let's break down the problem step by step:
a) Finding the formula connecting P, q, and R:
We can write the formula as P = k * q * R², where k is a constant. Now, we can substitute the given values of P, q, and R into the formula to find the value of k.
For the first set of values (P = 6, q = 12, R = 4):
6 = k * 12 * 4²
6 = k * 12 * 16
6 = k * 192
k = 6 / 192
k = 1/32
Now, we have the formula connecting P, q, and R as P = (1/32) * q * R².
b) Finding the value of P when q = 15 and R = 25:
Using the formula P = (1/32) * q * R², we substitute q = 15 and R = 25:
P = (1/32) * 15 * 25²
P = (1/32) * 15 * 625
P = 9375 / 32
P ≈ 292.97
Therefore, when q = 15 and R = 25, P is approximately equal to 292.97.
c) Finding the value of R when q = 4 and P = 2.5:
Using the formula P = (1/32) * q * R², we substitute q = 4 and P = 2.5:
2.5 = (1/32) * 4 * R²
2.5 = (1/2) * R²
R² = 2.5 * 2
R² = 5
R = √5
R ≈ 2.236
Therefore, when q = 4 and P = 2.5, R is approximately equal to 2.236.
To find the formula connecting P, q, and R, we can express the relationship using proportionality. We are given that P varies partly as q and partly as the square of R.
a) Let's first express the proportionality relationship when P = 6, q = 12, and R = 4:
P ∝ q.R^2
Substituting the given values:
6 ∝ 12.4^2
Simplifying further:
6 ∝ 12.16
To find the constant of proportionality, divide both sides of the equation by 12:
6/12 = 12.16/12
1/2 = 16/12
Now we can express the formula connecting P, q, and R:
P = (1/2) × q × R^2
b) To find the value of P when q = 15 and R = 25, we can simply substitute these values into the formula:
P = (1/2) × 15 × 25^2
P = (1/2) × 15 × 625
P = 7,812.5
Therefore, when q = 15 and R = 25, P = 7,812.5.
c) To find the value of R when q = 4 and P = 2.5, we rearrange the formula to isolate R:
P = (1/2) × q × R^2
Substituting the given values:
2.5 = (1/2) × 4 × R^2
2.5 = 2R^2
Dividing by 2:
1.25 = R^2
Taking the square root of both sides gives us two possible solutions:
R = ±√1.25
Therefore, when q = 4 and P = 2.5, the value of R can be either √1.25 or -√1.25.