1. **State the problem:** We need to find a real root of the cubic function $$f(x) = 9.34 - 2.19x + 16.3x^2 - 3.7x^3.$$ This means solving for $x$ such that $f(x) = 0$. 2. **Rewrite the equation:**$$9.34 - 2.19x + 16.3x^2 - 3.7x^3 = 0.$$ 3. **Analyze the polynomial:** This is a cubic equation; exact roots are typically found using methods such as the Rational Root Theorem or numerical methods. The coefficients are decimals, so numerical approximation methods like Newton's method or graphing are appropriate. 4. **Estimate bounds for the root:** We can test values to find sign changes, indicating roots. Evaluate at $x=1$: $$f(1) = 9.34 - 2.19(1) + 16.3(1)^2 - 3.7(1)^3 = 9.34 - 2.19 + 16.3 - 3.7 = 19.75 > 0.$$ Evaluate at $x=2$: $$f(2) = 9.34 - 2.19(2) + 16.3(4) - 3.7(8) = 9.34 - 4.38 + 65.2 - 29.6 = 40.56 > 0.$$ Evaluate at $x=3$: $$f(3) = 9.34 - 2.19(3) + 16.3(9) - 3.7(27) = 9.34 - 6.57 + 146.7 - 99.9 = 49.57 > 0.$$ Evaluate at $x=4$: $$f(4) = 9.34 - 2.19(4) + 16.3(16) - 3.7(64) = 9.34 - 8.76 + 260.8 - 236.8 = 24.58 > 0.$$ Evaluate at $x=5$: $$f(5) = 9.34 - 2.19(5) + 16.3(25) - 3.7(125) = 9.34 - 10.95 + 407.5 - 462.5 = -56.61 < 0.$$ There is a sign change between $x=4$ and $x=5$, so a root lies in $(4,5)$. 5. **Use bisection method in $(4,5)$:** Midpoint $x=4.5$ $$f(4.5) = 9.34 - 2.19(4.5) + 16.3(20.25) - 3.7(91.125) = 9.34 - 9.855 + 329.175 - 337.763 = -9.103 < 0.$$ Now root is in $(4,4.5)$ since $f(4)>0$ and $f(4.5)<0$. Midpoint $x=4.25$ $$f(4.25) = 9.34 - 2.19(4.25) + 16.3(18.0625) - 3.7(76.766) = 9.34 - 9.307 + 294.819 - 284.010 = 10.842 > 0.$$ Root is in $(4.25,4.5)$. Midpoint $x=4.375$ $$f(4.375) = 9.34 - 2.19(4.375) + 16.3(19.14) - 3.7(83.74) = 9.34 - 9.576 + 311.58 - 309.91 = 1.43 > 0.$$ Root is in $(4.375,4.5)$. Midpoint $x=4.4375$ $$f(4.4375) = 9.34 - 2.19(4.4375) + 16.3(19.69) - 3.7(87.44) = 9.34 - 9.71 + 320.95 - 323.53 = -3.96 < 0.$$ Root is in $(4.375,4.4375)$. Midpoint $x=4.40625$ $$f(4.40625) ext{ approx } 9.34 - 9.643 + 316.24 - 316.56 = -0.624 < 0.$$ Midpoint $x=4.390625$ $$f(4.390625) ext{ approx } 9.34 - 9.609 + 313.9 - 313.11 = 0.401 > 0.$$ 6. **Approximate root:** Between $4.390625$ ($f>0$) and $4.40625$ ($f<0$). The root is approximately $$x \\approx 4.4$$. **Answer:** A real root of $f(x)$ is approximately $$\boxed{4.4}.$$