Math Snap

STEP 1

1. We are given a definite integral to evaluate.
2. The function to integrate is $\frac{6t - 9}{\sqrt{t}}$.
3. The limits of integration are from $t = 1$ to $t = 16$.
4. The problem requires the use of basic integration techniques and the Fundamental Theorem of Calculus.

STEP 2

1. Simplify the integrand.
2. Integrate the simplified function.
3. Evaluate the definite integral using the limits of integration.

STEP 3

Simplify the integrand $\frac{6t - 9}{\sqrt{t}}$.
Rewrite the expression as two separate terms:
$$\frac{6t}{\sqrt{t}} - \frac{9}{\sqrt{t}}$$ Simplify each term:
$$6t^{1/2} - 9t^{-1/2}$$

STEP 4

Integrate the simplified function $6t^{1/2} - 9t^{-1/2}$.
The antiderivative of $6t^{1/2}$ is:
$$\int 6t^{1/2} \, dt = 6 \cdot \frac{t^{3/2}}{3/2} = 4t^{3/2}$$ The antiderivative of $-9t^{-1/2}$ is:
$$\int -9t^{-1/2} \, dt = -9 \cdot \frac{t^{1/2}}{1/2} = -18t^{1/2}$$ Combine the antiderivatives:
$$4t^{3/2} - 18t^{1/2}$$

Evaluate the definite integral from $t = 1$ to $t = 16$.
Substitute the upper limit $t = 16$ into the antiderivative:
$$4(16)^{3/2} - 18(16)^{1/2}$$ Calculate each term:
$$4 \times 64 - 18 \times 4 = 256 - 72 = 184$$ Substitute the lower limit $t = 1$ into the antiderivative:
$$4(1)^{3/2} - 18(1)^{1/2} = 4 \times 1 - 18 \times 1 = 4 - 18 = -14$$ Subtract the lower limit evaluation from the upper limit evaluation:
$$184 - (-14) = 184 + 14 = 198$$ The value of the definite integral is:
$$\boxed{198}$$