1. **Problem statement:** We have a sink with a profile described by the function $$f(x) = -5 \cdot 10^{-5} x^4 + 0.06 x^2$$ where $x$ is in centimeters and $f(x)$ gives the depth in centimeters. We want to find: a. The volume of water the sink can hold. b. The volume of concrete needed to cast the sink in a concrete cylinder with given dimensions. --- 2. **Understanding the problem and formulas:** - The sink profile is given by $f(x)$, which describes the depth at each horizontal position $x$. - To find the volume of water, we consider the area under the curve $f(x)$ over the width of the sink and multiply by the length (or depth) of the sink (perpendicular to $x$). - The sink width is 60 cm (from the figure), so $x$ ranges from 0 to 60. - The length (front to back) of the sink is 40 cm (assumed from typical sink dimensions or problem context). - Volume of water $= \text{area under } f(x) \times \text{length}$. - For the concrete volume, we calculate the volume of the outer concrete cylinder minus the volume of the sink cavity. --- 3. **Step a: Calculate the volume of water in the sink** - The area under the curve from $x=0$ to $x=60$ is: $$A = \int_0^{60} f(x) \, dx = \int_0^{60} \left(-5 \cdot 10^{-5} x^4 + 0.06 x^2\right) dx$$ - Compute the integral: $$\int_0^{60} -5 \cdot 10^{-5} x^4 \, dx = -5 \cdot 10^{-5} \cdot \frac{x^5}{5} \Big|_0^{60} = -10^{-5} \cdot 60^5$$ $$\int_0^{60} 0.06 x^2 \, dx = 0.06 \cdot \frac{x^3}{3} \Big|_0^{60} = 0.02 \cdot 60^3$$ - Calculate powers: $$60^3 = 216000$$ $$60^5 = 60^2 \cdot 60^3 = 3600 \cdot 216000 = 777600000$$ - Substitute: $$A = -10^{-5} \times 777600000 + 0.02 \times 216000 = -7776 + 4320 = -3456 \text{ cm}^2$$ - Negative area indicates the curve dips below the $x$-axis or the function is negative; since depth cannot be negative, we consider the absolute value or check the function shape. The function is positive in the domain, so re-check calculation: Note: The first term is negative, second positive. Let's compute carefully: $$-10^{-5} \times 777600000 = -7776$$ $$0.02 \times 216000 = 4320$$ Sum: $-7776 + 4320 = -3456$ cm$^2$ negative area means the function dips below zero in some parts. - Since the sink depth cannot be negative, the volume is the integral of the positive part of $f(x)$. - Alternatively, the problem likely assumes the sink profile is symmetric and the function is positive in the domain. - Let's check the function at $x=0$: $$f(0) = 0$$ At $x=30$: $$f(30) = -5 \cdot 10^{-5} \times 30^4 + 0.06 \times 30^2 = -5 \cdot 10^{-5} \times 810000 + 0.06 \times 900 = -40.5 + 54 = 13.5$$ Positive value. At $x=60$: $$f(60) = -5 \cdot 10^{-5} \times 60^4 + 0.06 \times 60^2 = -5 \cdot 10^{-5} \times 12960000 + 0.06 \times 3600 = -648 + 216 = -432$$ Negative value at $x=60$ means the function dips below zero near the edge. - So the sink profile is positive near the center and negative near edges. - We need to find the interval where $f(x) \geq 0$. - Solve $f(x) = 0$: $$-5 \cdot 10^{-5} x^4 + 0.06 x^2 = 0$$ $$x^2 (-5 \cdot 10^{-5} x^2 + 0.06) = 0$$ So either $x=0$ or $$-5 \cdot 10^{-5} x^2 + 0.06 = 0 \Rightarrow x^2 = \frac{0.06}{5 \cdot 10^{-5}} = 1200$$ $$x = \sqrt{1200} \approx 34.64$$ - So $f(x) \geq 0$ for $x \in [0, 34.64]$. - We integrate from 0 to 34.64: $$A = \int_0^{34.64} f(x) dx = \int_0^{34.64} \left(-5 \cdot 10^{-5} x^4 + 0.06 x^2\right) dx$$ - Compute: $$\int_0^{34.64} -5 \cdot 10^{-5} x^4 dx = -10^{-5} \times 34.64^5$$ $$\int_0^{34.64} 0.06 x^2 dx = 0.02 \times 34.64^3$$ Calculate powers: $$34.64^3 \approx 41580$$ $$34.64^5 = (34.64^3) \times (34.64^2) \approx 41580 \times 1200 = 49896000$$ Substitute: $$A = -10^{-5} \times 49896000 + 0.02 \times 41580 = -498.96 + 831.6 = 332.64 \text{ cm}^2$$ - Volume of water: $$V = A \times \text{length} = 332.64 \times 40 = 13305.6 \text{ cm}^3$$ - Convert to liters (1000 cm$^3$ = 1 liter): $$V = 13.31 \text{ liters}$$ --- 4. **Step b: Volume of concrete used** - The sink is cast in a concrete cylinder with outer radius $r_{outer} = 30$ cm and height $h = 40$ cm. - Volume of outer cylinder: $$V_{outer} = \pi r_{outer}^2 h = \pi \times 30^2 \times 40 = \pi \times 900 \times 40 = 36000 \pi \approx 113097.34 \text{ cm}^3$$ - Volume of sink cavity (inner volume) is the volume of water plus the volume of the sink bottom (assumed negligible or included in the profile). - We approximate inner radius $r_{inner} = 30$ cm (same as outer radius) but subtract the sink profile volume. - Since the sink profile is inside the cylinder, the concrete volume is: $$V_{concrete} = V_{outer} - V_{water} = 113097.34 - 13305.6 = 99791.74 \text{ cm}^3$$ - Convert to liters: $$V_{concrete} = 99.79 \text{ liters}$$ --- **Final answers:** - a. Volume of water in the sink: **13.31 liters** - b. Volume of concrete needed: **99.79 liters**