1. **Problem statement:** We have a cable curve described by a quadratic function $f(x) = a x^2 + b x + c$ between two supports A and B. Given: - Horizontal distance between A and B is 50 m. - Both supports A and B are at height 15 m. - The tangent at point B has slope 0.5. Tasks: (a) Choose a coordinate system and find $a$, $b$, and $c$ such that the tangent at B has slope 0.5. (b) Find the coordinates of the lowest point $T$ of the cable. (c) Find the point where the sag $d$ (vertical dip) is greatest. 2. **Choosing a coordinate system:** Place the origin at support A: $x=0$ at A, so $f(0) = 15$ m. Then B is at $x=50$, $f(50) = 15$ m. 3. **Using the conditions:** - At A: $f(0) = c = 15$ - At B: $f(50) = a imes 50^2 + b imes 50 + c = 15$ - The slope at B is $f'(50) = 2 a imes 50 + b = 0.5$ 4. **Write equations:** From $f(0) = 15$: $$ c = 15 $$ From $f(50) = 15$: $$ 2500 a + 50 b + 15 = 15 $$ Simplify: $$ 2500 a + 50 b = 0 $$ From slope at B: $$ f'(x) = 2 a x + b $$ $$ f'(50) = 100 a + b = 0.5 $$ 5. **Solve the system:** From $2500 a + 50 b = 0$, divide both sides by 50: $$ \cancel{50} 50 a + \cancel{50} b = 0 \Rightarrow 50 a + b = 0 $$ We have two equations: $$ 50 a + b = 0 $$ $$ 100 a + b = 0.5 $$ Subtract the first from the second: $$ (100 a + b) - (50 a + b) = 0.5 - 0 $$ $$ 50 a = 0.5 $$ $$ a = \frac{0.5}{50} = 0.01 $$ Use $50 a + b = 0$: $$ 50 \times 0.01 + b = 0 $$ $$ 0.5 + b = 0 $$ $$ b = -0.5 $$ 6. **Final coefficients:** $$ a = 0.01, \quad b = -0.5, \quad c = 15 $$ 7. **Find the lowest point $T$:** The vertex of the parabola is at $$ x_T = -\frac{b}{2a} = -\frac{-0.5}{2 \times 0.01} = \frac{0.5}{0.02} = 25 $$ Calculate $y_T$: $$ f(25) = 0.01 \times 25^2 - 0.5 \times 25 + 15 = 0.01 \times 625 - 12.5 + 15 = 6.25 - 12.5 + 15 = 8.75 $$ So the lowest point is $$ T = (25, 8.75) $$ 8. **Point of maximum sag $d$:** Sag $d$ is the vertical difference between the supports and the cable lowest point. Since the supports are at 15 m and the lowest point is at 8.75 m, the sag is $$ d = 15 - 8.75 = 6.25 $$ The sag is greatest at the lowest point $T$. **Final answers:** - (a) $a=0.01$, $b=-0.5$, $c=15$ - (b) Lowest point $T = (25, 8.75)$ - (c) Maximum sag $d=6.25$ m at $x=25$