# Differential Calculus

## How Far An Object Has Fallen If Its Velocity Is 80 Feet Per Second

**Problem**

The formula $v = \sqrt{2gh}$ give the velocity, in feet per second, of an object when it falls *h* feet accelerated by gravity *g*, in feet per second squared. If *g* is approximately 32 feet per second squared, find how far an object has fallen if its velocity is 80 feet per second.

A. 80 feet | C. 70 feet |

B. 100 feet | D. 90 feet |

## Volume of Inflating Spherical Balloon as a Function of Time

**Problem**

A meteorologist is inflating a spherical balloon with a helium gas. If the radius of a balloon is changing at a rate of 1.5 cm/sec., express the volume *V* of the balloon as a function of time *t* (in seconds). Hint: Use composite function relationship *V*_{sphere} = 4/3 π*r*^{3} as a function of *x* (radius), and *x* (radius) as a function of *t* (time).

A. V(t) = 5/2 πt^{3} |
C. V(t) = 9/2 πt^{3} |

B. V(t) = 7/2 πt^{3} |
D. V(t) = 3/2 πt^{3} |

## Smallest Triangular Portion From A Square Lot

**Problem**

A farmer owned a square field measuring exactly 2261 m on each side. 1898 m from one corner and 1009 m from an adjacent corner stands Narra tree. A neighbor offered to purchase a triangular portion of the field stipulating that a fence should be erected in a straight line from one side of the field to an adjacent side so that the Narra tree was part of the fence. The farmer accepted the offer but made sure that the triangular portion was a minimum area. What was the area of the field the neighbor received and how long was the fence? Hint: Use the Cosine Law.

A. A = 972,325 m^{2} and L = 2,236 m |

B. A = 950,160 m^{2} and L = 2,122 m |

C. A = 946,350 m^{2} and L = 2,495 m |

D. A = 939,120 m^{2} and L = 2,018 m |

## Dimensions of the Lot for a Given Cost of Fencing

**Problem**

A rectangular waterfront lot has a perimeter of 1000 feet. To create a sense of privacy, the lot owner decides to fence along three sides excluding the sides that fronts the water. An expensive fencing along the lot’s front length costs Php25 per foot, and an inexpensive fencing along two side widths costs only Php5 per foot. The total cost of the fencing along all three sides comes to Php9500. What is the lot’s dimensions?

A. 300 feet by 100 feet | C. 400 feet by 200 feet |

B. 400 feet by 100 feet | D. 300 feet by 200 feet |

## Rate of Change of Volume of Sand in Conical Shape

**Problem**

A conveyor is dispersing sands which forms into a conical pile whose height is approximately 4/3 of its base radius. Determine how fast the volume of the conical sand is changing when the radius of the base is 3 feet, if the rate of change of the radius is 3 inches per minute.

A. 2π ft/min | C. 3π ft/min |

B. 4π ft/min | D. 5π ft/min |

## Velocity of Separation: How fast is the distance between two cars changing?

**Problem**

A Toyota Land Cruiser drives east from point *A* at 30 kph. Another car, Ford Expedition, starting from *B* at the same time, drives S30°W toward *A* at 60 kph. *B* is 30 km from *A*. How fast in kph is the distance between two cars changing after 30 minutes? Hint: Use the Cosine Law.

A. 70 kph | C. 55 kph |

B. 80 kph | D. 60 kph |

## Find y’ if x = 2 arccos 2t and y = 4 arcsin 2t

**Problem**

Find *y’* if *x* = 2 arccos 2*t* and *y* = 4 arcsin 2*t*.

A. 2 | C. 4 |

B. -2 | D. -4 |

## Equation of the Diameter of Parabola Bisecting Parallel Chords of Given Slope

**Problem**

A parabola has an equation of *y*^{2} = 8*x*. Find the equation of the diameter of the parabola, which bisect chords parallel to the line *x* – *y* = 4.

A. y = 2 |
C. y = 4 |

B. y = 3 |
D. y = 1 |

## Slope of a Curve of Given Parametric Equations

**Problem**

A point moves in the plane according to equations *x* = *t*^{2} + 2*t* and *y* = 2*t*^{3} - 6*t*. Find *dy*/*dx* when *t* = 0, 2, 5.

A. -3, -3, -12 | C. 3, 3, 12 |

B. 3, -3, 12 | D. -3, 3, 12 |