The distance a wave runner is from a dock on a shoreline is represented by the function f(d)=49+d2−−−−−−√

Question

, where d is the distance the wave runner has traveled since noon. The distance the wave runner has traveled is represented by the function g(t)=6t
, where t is the time in hours the wave runner has traveled. What is (f∘g)(t)
and what does it mean? What is the domain of (f∘g)(t)
?(1 point)
Responses

(f∘g)(t)=649+t2−−−−−−√
represents the distance the wave runner travels. The domain is t≥0
.
open paren f circ g close paren times t is equal to 6 square root of 49 plus t squared end root represents the distance the wave runner travels. The domain is t is greater than or equal to 0 .

(f∘g)(t)=49+36t2−−−−−−−−√
represents the distance the wave runner is from the dock after t hours. The domain is (−∞, ∞)
.
open paren f circ g close paren times t is equal to square root of 49 plus 36 t squared end root represents the distance the wave runner is from the dock after t hours. The domain is open paren negative infinity comma infinity close paren .

(f∘g)(t)=649+t2−−−−−−√
represents the distance the wave runner travels. The domain is (−∞, ∞)
.
open paren f circ g close paren times t is equal to 6 square root of 49 plus t squared end root represents the distance the wave runner travels. The domain is open paren negative infinity comma infinity close paren .

(f∘g)(t)=49+36t2−−−−−−−−√
represents the distance the wave runner is from the dock after t hours. The domain is t≥0
.

Answer 1

To find \((f \circ g)(t)\), we need to substitute \(g(t)\) into \(f(d)\).

The function \(g(t)\) is given by: \[ g(t) = 6t. \] The function \(f(d)\) is given by: \[ f(d) = \sqrt{49 + d^2}. \]

Now we substitute \(g(t)\) into \(f(d)\): \[ (f \circ g)(t) = f(g(t)) = f(6t) = \sqrt{49 + (6t)^2} = \sqrt{49 + 36t^2}. \]

This means that \((f \circ g)(t)\) represents the distance the wave runner is from the dock after \(t\) hours.

Now, let's determine the domain of \((f \circ g)(t)\).

Since \(t\) represents the time in hours that the wave runner has traveled, it is restricted to non-negative values: \[ t \geq 0 \] because time cannot be negative.

Thus, the domain of \((f \circ g)(t)\) is: \[ [0, \infty) \]

Now, we can match this to the provided multiple choice responses. The correct response should be:

\((f\circ g)(t)=\sqrt{49+36t^2}\) represents the distance the wave runner is from the dock after \(t\) hours. The domain is \(t \geq 0\).