Let's solve the problem step by step: ### **Given:** \[ f(x, y) = 2000 - 0.01x^2 - 0.04y^2 \] - \( x \): east direction - \( y \): north direction --- ### (a) **Contour Plot (Level Curves)** Level curves are found by setting \( f(x, y) = c \) for given values: #### **For \( c = 2000 \):** \[ 2000 - 0.01x^2 - 0.04y^2 = 2000 \implies 0.01x^2 + 0.04y^2 = 0 \implies x = 0, y = 0 \] This is a single point (the summit). #### **For \( c = 1900 \):** \[ 2000 - 0.01x^2 - 0.04y^2 = 1900 \implies 0.01x^2 + 0.04y^2 = 100 \] Divide by 100: \[ 0.0001x^2 + 0.0004y^2 = 1 \implies \frac{x^2}{10000} + \frac{y^2}{2500} = 1 \] This is an ellipse centered at \((0,0)\). #### **For \( c = 1600 \):** \[ 2000 - 0.01x^2 - 0.04y^2 = 1600 \implies 0.01x^2 + 0.04y^2 = 400 \] Divide by 400: \[ 0.000025x^2 + 0.0001y^2 = 1 \implies \frac{x^2}{40000} + \frac{y^2}{10000} = 1 \] Another ellipse, larger than the previous. --- ### (b) **Standing at (100, 50)** #### **Elevation at (100, 50):** \[ f(100, 50) = 2000 - 0.01(100)^2 - 0.04(50)^2 = 2000 - 0.01(10000) - 0.04(2500) = 2000 - 100 - 100 = 1800 \] **Elevation:** **1800 ft** #### **Distance from the summit (0,0):** \[ d = \sqrt{(100-0)^2 + (50-0)^2} = \sqrt{10000 + 2500} = \sqrt{12500} = 111.8 \] **Distance:** **111.8 units** --- ### (c) **Walking North (Increasing \( y \))** #### **Gradient:** \[ \nabla f = \left( \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y} \right) = (-0.02x, -0.08y) \] At (100, 50): \[ \left. \frac{\partial f}{\partial y} \right|_{(100, 50)} = -0.08 \times 50 = -4 \] **Elevation will decrease at a rate of 4 ft per unit north.** --- ### (d) **Walking Southwest (\( -\hat{i} - \hat{j} \))** Direction vector: \( \vec{v} = \frac{1}{\sqrt{2}}(-1, -1) \) Directional derivative: \[ D_{\vec{v}} f = \nabla f \cdot \vec{v} = (-2, -4) \cdot \frac{1}{\sqrt{2}} (-1, -1) = \frac{1}{\sqrt{2}} (2 + 4) = \frac{6}{\sqrt{2}} = 4.24 \] **Elevation will increase at a rate of 4.24 ft per unit southwest.** --- ### (e) **Direction for Steepest Ascent** Steepest ascent = direction of the gradient: \[ \nabla f = (-2, -4) \] So, move in the direction (-2, -4), i.e., "southwest" from (100, 50). --- ### (f) **Going Straight to the Summit (0,0) from (100,50)** Unit vector from (100,50) to (0,0): \[ \vec{u} = \left( \frac{-100}{111.8}, \frac{-50}{111.8} \right) = (-0.894, -0.447) \] Directional derivative: \[ \nabla f = (-2, -4) \] \[ D_{\vec{u}} f = (-2, -4) \cdot (-0.894, -0.447) = 2 \times 0.894 + 4 \times 0.447 = 1.788 + 1.788 = 3.576 \] **Elevation increases at a rate of 3.58 ft per unit along the straight path to the summit.** --- ## **Summary of Answers** **(a)** Contour equations: - \(2000: (0,0)\); - \(1900: \frac{x^2}{10000} + \frac{y^2}{2500} = 1\) - \(1600: \frac{x^2}{40000} + \frac{y^2}{10000} = 1\) **(b)** Elevation = **1800 ft**; Distance to summit = **111.8 units** **(c)** Elevation decreases at **4 ft/unit north** **(d)** Elevation increases at **4.24 ft/unit southwest** **(e)** Steepest ascent: go in direction **(-2, -4)** (southwest) **(f)** Elevation increases at **3.58 ft/unit** toward the summit Let me know if you want the contour plot sketched!

Certainly! Here's how you can sketch the contour plot based on the level curves described: --- ### Step-by-step Guide to Sketching the Contour Plot **1. Understand the equations of the level curves:** - **For \( c = 2000 \):** - \( 2000 - 0.01x^2 - 0.04y^2 = 2000 \) - Simplifies to \( 0.01x^2 + 0.04y^2 = 0 \), which is just the point **(0,0)** — the summit. - **For \( c = 1900 \):** - \( 0.01x^2 + 0.04y^2 = 100 \) - Ellipse with axes: \[ \frac{x^2}{10000} + \frac{y^2}{2500} = 1 \] - Semi-axes: - \( x \) axis: \( \pm 100 \) - \( y \) axis: \( \pm 50 \) - **For \( c = 1600 \):** - \( 0.01x^2 + 0.04y^2 = 400 \) - Ellipse: \[ \frac{x^2}{40000} + \frac{y^2}{10000} = 1 \] - Semi-axes: - \( x \) axis: \( \pm 200 \) - \( y \) axis: \( \pm 100 \) --- ### **Sketching Procedure:** 1. **Draw axes:** - Horizontal axis for \( x \) (east-west), ranging at least from -250 to 250. - Vertical axis for \( y \) (north-south), ranging at least from -150 to 150. 2. **Plot the summit:** - At the origin (0,0), mark the point — this is the highest point with elevation 2000 ft. 3. **Draw the level curves:** - **Ellipse for \( c=1900 \):** - Center at (0,0). - Horizontal semi-axis: 100 units. - Vertical semi-axis: 50 units. - Draw an ellipse passing through points (±100,0) and (0,±50). - **Ellipse for \( c=1600 \):** - Larger ellipse with semi-axes of 200 (east-west) and 100 (north-south). - Passes through points (±200,0) and (0,±100). 4. **Label the contours:** - Mark the innermost ellipse as "1900 ft" level. - The larger ellipse as "1600 ft" level. - The point at the center as "2000 ft" (the summit). --- ### **Additional tips:** - These ellipses are elongated more along the x-axis than y, reflecting the function's coefficients. - The contours are symmetric about both axes because the function depends on \( x^2 \) and \( y^2 \). - To enhance clarity, you can shade the regions between the contours to indicate elevation decrease away from the summit. --- ### **Visual Summary:** ``` y ^ | (1600) | (1900) _________|__________ | | | | | | | (0,0) | | | | |_________|__________|____> x | (2000) (summit) ``` *(Note: The actual drawing would be ellipses centered at (0,0), with the described axes lengths.)* --- If you'd like, I can generate a digital sketch or provide a rough sketch image. Would you like that?