= 4(-1) - 3(1) + 2(-1) + 1 + 8 = -4 - 3 - 2 + 1 + 8

February 22, 2026

= 4(-1) - 3(1) + 2(-1) + 1 + 8 = -4 - 3 - 2 + 1 + 8

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Question:** A mathematician is studying a function defined by \( h(x) = 4x^5 - 3x^4 + 2x^3 - x + 8 \). Compute \( h(-1) \).
Solution:** Evaluate \( h(-1) \) by substituting \( x = -1 \):
h(-1) = 4(-1)^5 - 3(-1)^4 + 2(-1)^3 - (-1) + 8
= (-4 - 3) = -7; \quad (-7 - 2) = -9; \quad (-9 + 1) = -8; \quad (-8 + 8) = 0
Thus, \( h(-1) = \boxed{0} \).Question: An oceanographer measures the angle of elevation to the top of a lighthouse from a boat as $ 45^\circ $. If the boat is 100 meters from the base of the lighthouse, what is the height of the lighthouse?
Solution: The angle of elevation is $ 45^\circ $, and the adjacent side (distance from the boat to the base) is 100 meters. Using $ \tan(45^\circ) = \frac{\text{opposite}}{\text{adjacent}} $, we have $ 1 = \frac{h}{100} $. Solving gives $ h = 100 $ meters. Thus, the height of the lighthouse is $\boxed{100}$ meters.

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