#### 2278.125**Question:** A geophysicist is analyzing the cross-sectional area of a circular seismic sensor with a diameter of 10 cm. If a rectangular area of 6 cm by 8 cm is subtracted from the circle, what is the remaining area of the cross-section? Express your answer in terms of \(\pi\).

["Title: How to Calculate the Remaining Cross-Sectional Area After Removing a Rectangle from a Circular Sensor", "When analyzing seismic sensor designs, geophysicists often need to understand the effective cross-sectional area after modifying the shape—such as removing a rectangular component from a circular sensor. This article explores a practical problem involving a circular seismic sensor with a diameter of 10 cm, from which a 6 cm by 8 cm rectangle has been subtracted, and calculates the remaining area in terms of (\pi).", "---", "### Step 1: Determine the Area of the Circular Sensor", "The sensor is circular with a diameter of 10 cm, so its radius (r) is:", "[\nr = \frac{10}{2} = 5 \ ext{ cm}\n]", "The area (A_{\ ext{circle}}) of a circle is given by the formula:", "[\nA_{\ ext{circle}} = \pi r^2 = \pi (5)^2 = 25\pi \ ext{ cm}^2\n]", "---", "### Step 2: Compute the Area of the Subtracted Rectangle", "The rectangular region removed measures 6 cm by 8 cm. Its area (A_{\ ext{rect}}) is:", "[\nA_{\ ext{rect}} = 6 \ imes 8 = 48 \ ext{ cm}^2\n]", "---", "### Step 3: Subtract to Find the Remaining Cross-Sectional Area", "The remaining area (A_{\ ext{remaining}}) is the area of the full circle minus the area of the subtracted rectangle:", "[\nA_{\ ext{remaining}} = A_{\ ext{circle}} - A_{\ ext{rect}} = 25\pi - 48 \ ext{ cm}^2\n]", "---", "### Final Answer:", "[\n\boxed{25\pi - 48} \ ext{ cm}^2\n]", "This expression gives the remaining cross-sectional area of the seismic sensor in terms of (\pi), illustrating the impact of structural modifications on sensor geometry. Understanding such calculations helps optimize sensor design for improved seismic wave detection and signal clarity.", "---", "Keywords: circular seismic sensor, cross-sectional area, geophysics, rectangle area subtracted, 25\pi - 48, seismic sensor design, area calculation.", "Meta Description:\nDiscover how a geophysical analysis calculates the remaining area after removing a 6 cm × 8 cm rectangle from a 10 cm diameter circular sensor. The answer is (25\pi - 48) cm²—ideal for sensor design and engineering applications."]