Suppose you are climbing a hill whose shape is given by the equationz = 1,300 − 0.005x2 − 0.01y2,where x, y, and z are measured in meters, and you are standing at a point with coordinates (80, 80, 1204). The positive x-axis points east and the positive y-axis points north. ### Problem StatementSuppose you are climbing a hill whose shape is given by the equation \( z = 1,300 - 0.05x^2 - 0.01y^2 \), where \( x, y, \) and \( z \) are measured in meters, and you are standing at a point with coordinates \( (80, 80, 1,204) \). The positive x-axis points east and the positive y-axis points north.### Questions#### (a) If you walk due south, will you start to ascend or descend?- Options: - ⬜ ascend - ⬜ descend- **At what rate (in vertical meters per horizontal meter)?** \[ 1.6 \] vertical meters per horizontal meter#### (b) If you walk northwest, will you start to ascend or descend?- Options: - ⬜ ascend - ⬜ descend- **At what rate (in vertical meters per horizontal meter)?** (Round your answer to two decimal places.) \[ \_\_\_\_ \] vertical meters per horizontal meter#### (c) In which direction is the slope largest?- **Direction:** \[ \_\_\_\_ \]- **What is the rate of ascent (in vertical meters per horizontal meter) in that direction?** (Round your answer to two decimal places.) \[ \_\_\_\_ \] vertical meters per horizontal meter- **At what angle (in degrees) above the horizontal does the path in that direction begin?** (Round your answer to two decimal places.) \[ \_\_\_\_ \] °### ExplanationThe problem involves determining changes in elevation (ascent or descent) based on direction, using a quadratic equation that represents the shape of a hill. The given coordinates are used to calculate the rate of ascent or descent and the direction of the maximum slope using gradient analysis.

Suppose you are climbing a hill whose shape is given by the equation z=1300-0.005x2-0.01y2, where x,y, and z are measured in meters, and you are standing at a point with coordinates 80, 80, 1204. The positive x-axis points east and the positive y-axis points north. We have find the required values.a) z=1300-0.005x2-0.01y2 you walk due to south, the directional derivative in the direction of -j. This is the negative of partial derivative y. Since zy=-0.02y, the value is equal to 0.0280=1.6. This is positive, this means that the hill slopes upward in this direction. Therefore, if you walk due to south, you will start to ascend. b) The unit vector in the direction of northwest is given by u=12-1, 1. Duz80, 80=∇z80, 80.u=-0.0180, -0.0280.12-1, 1=12-0.8, -1.6-1, 1=120.8-1.6Duz80, 80=-0.82≈-0.57 Since this is negative, the hill slopes downward in this direction. Therefore, if you walk due to northwest, you will start to descend. The rate is 0.57 vertical meters per horizontal meters.c) The slope is largest in the direction of gradient. ∇z80, 80=∂z∂x80, 80, ∂z∂y80, 80=-0.0180, -0.0280∇z80, 80=-0.8, -1.6 The slope is largest in the direction ∇z80, 80=-0.8, -1.6. The rate of ascent, rate=-0.8, -1.6=-0.82+-1.62=0.64+2.56rate≈1.79 The rate of ascent is 1.79 vertical meters per horizontal meters. Angle above the horizontal in this path, θ=tan-11.788854=60.79° The angle above the horizontal in this path is 60.79°.Final answer: a) If you walk due to south, you will start to ascend. b) If you walk due to northwest, you will start to descend. The rate is 0.57 vertical meters per horizontal meters. c) The slope is largest in the direction ∇z80, 80=-0.8, -1.6. The rate of ascent is 1.79 vertical meters per horizontal meters. The angle above the horizontal in this path is 60.79°.