Question

H is inversely proportional to p

P is directly proportional to root of t
Given that H equals 10 and T equals 14r when p equals six
Find a formula of h in terms of t

Answer 1

To find the formula for H in terms of t, we use the inversely proportional relationship between H and p, and the directly proportional relationship between p and √(t). The relationship can be expressed as H = k/p and p = c√(t), where k and c are constants. With the given values, we find k = 60, and the formula for H in terms of t would be H = 60/(c√(t)), with c being an unknown constant without additional information.

Step-by-step explanation:

The student is asking to find a formula for H in terms of t, given that H is inversely proportional to p and p is directly proportional to the square root of t.

To find the formula, we need to consider both of these relationships and apply them to the given values: H = 10 when p = 6, and t is represented by 14r.

Since H is inversely proportional to p, we can write this relationship as H = k/p, where k is a constant of proportionality. Similarly, since p is directly proportional to √(t), we can write p = c√(t), where c is another constant of proportionality.

To find the value of k, we use the given that H equals 10 when p equals 6.

Therefore, k = H * p = 10 * 6 = 60.

Now we can express H in terms of p: H = 60/p. Since p is proportional to √(t), and given t = 14r, we can plug the value into the relationship for p: p = c√(14r)

To find c, we need the other given relationship for p and t.

However, we're not provided with a second condition to find the exact value for c.

Assuming we had such a condition, we would write p in terms of t, find c, and then substitute c√(t) for p in the equation H = 60/p to find H in terms of t.

Without this second condition, the formula for H in terms of t would be H = 60/(c√(t)), where c remains an unknown constant that would need to be determined from additional information.

Answer 2

To find a formula for H in terms of t, first find the constants of proportionality from given values, and then express H in terms of p and subsequently in terms of t by eliminating p.

Step-by-step explanation:

The question involves finding a formula for H in terms of t, where H is inversely proportional to p, and p is directly proportional to the square root of t. Given H = 10 when p = 6 and t = 14r, we can set up two proportional relationships:

• H Α 1/p
• p Α √t

We can express these proportionalities using constants of proportionality, k1 and k2:

• H = k1/p
• p = k2√t

Since we know H and p at specific values, we can use these to solve for the constants k1 and k2, then eliminate p to get H explicitly in terms of t:

1. For H when p = 6, H = 10: 10 = k1/6 → k1 = 60.
2. For p when t = 14r (given p = 6): 6 = k2√14r → k2 = 6/√14r.

Plugging k2 back into the equation for p in terms of t gives:

p = (6/√14r)√t

And substituting p back into the equation for H gives:

H = 60 / ((6/√14r)√t) = (60√14r) / (6√t) = 10√14r / √t