Instructions
It is a truth universally acknowledged that a student in possession of a good mind must be in want of a challenge. Pray, apply your knowledge of geometry and algebra to the following predicaments of country life. You are to show your work with diligence and present your final answer with the utmost clarity. A calculator may be of assistance in your more complex endeavours, but do not neglect your reasoning.
Part I: Journeys and Distances
In which the shortest path is determined with the aid of the Pythagorean Theorem.
- 1. A gentleman wishes to ride directly from his estate to a neighbouring manor. To travel by road, he must go 8 miles south and then 15 miles east. What is the distance of the direct, straight-line path he might take across the open fields? (This forms a right-angled triangle).
- 2. A ladder, 25 feet in length, is leant against a great house to assist a painter. The base of the ladder is placed 7 feet away from the wall of the house. How high up the wall does the ladder reach? Please provide the height in feet.
Part II: The Design of Gardens and Grounds
In which special right triangles are employed to ensure elegance and order.
- 3. A landscape architect is designing a perfectly square garden plot. To ensure the corners are true right angles, she measures the diagonal path that cuts across the garden. If each side of the square garden is 30 feet, what is the length of this diagonal? Express your answer in simplest radical form. (Consider a 45-45-90 triangle).
- 4. A triangular section of a park is to be fenced for a new folly. The section has angles of 30, 60, and 90 degrees. The side opposite the 30-degree angle (the shortest side) measures 12 yards. What are the lengths of the other two sides (the side opposite the 60-degree angle and the hypotenuse)? Express your answers in simplest radical form.
Part III: On Ballrooms and Fields
In which the areas of various quadrilaterals are calculated for purposes of society and agriculture.
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5. The grand ballroom at Netherfield is a rectangle of 50 feet in length and 30 feet in width.
a) Write an expression for the area of the ballroom using variables L for length and W for width.
b) Calculate the total area of the ballroom floor in square feet. - 6. A farmer owns a trapezoidal field. The two parallel sides measure 200 yards and 300 yards. The perpendicular distance (the height) between these two sides is 90 yards. What is the total area of the field in square yards?
Part IV: The Economics of an Estate
In which formulae are rearranged and inequalities are reasoned with to manage affairs.
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7. The formula for the perimeter, P, of a rectangular field is P = 2L + 2W, where L is the length and W is the width. A landowner wishes to purchase fencing. He knows the perimeter he can afford is 320 yards, and the width of his field is 60 yards.
a) Rearrange the formula to solve for the length, L.
b) Using your new formula, calculate the length of the field. -
8. A lady is planning a small party. The cost of hiring musicians is a fixed fee of £10. The refreshments cost £2 per guest, g. Her budget for the affair must not exceed £40.
a) Write an inequality to represent this financial situation.
b) Solve the inequality to determine the maximum number of guests she can invite.
Answer Key
Pray, examine the solutions below to ascertain the correctness of your own calculations.
Part I: Journeys and Distances
1. Using the Pythagorean Theorem, a² + b² = c².
8² + 15² = c²
64 + 225 = c²
289 = c²
c = √289 = 17 miles. The direct path is 17 miles. (This is a Pythagorean triple: 8-15-17).
2. The ladder is the hypotenuse (c), and the distance from the wall is one leg (a). We must find the other leg (b).
a² + b² = c²
7² + b² = 25²
49 + b² = 625
b² = 625 - 49 = 576
b = √576 = 24 feet. The ladder reaches 24 feet up the wall.
Part II: The Design of Gardens and Grounds
3. In a 45-45-90 triangle, the hypotenuse is √2 times the length of a leg.
Hypotenuse = leg × √2
Hypotenuse = 30 × √2 = 30√2 feet.
4. In a 30-60-90 triangle, the sides are in the ratio x : x√3 : 2x, opposite the 30°, 60°, and 90° angles respectively.
The shortest side (opposite 30°) is given as 12 yards, so x = 12.
The side opposite 60° = x√3 = 12√3 yards.
The hypotenuse (opposite 90°) = 2x = 2(12) = 24 yards.
Part III: On Ballrooms and Fields
5.
a) The expression for the area is A = L × W.
b) A = 50 ft × 30 ft = 1500 square feet.
6. The formula for the area of a trapezoid is A = ½(b₁ + b₂)h.
A = ½(200 + 300) × 90
A = ½(500) × 90
A = 250 × 90 = 22,500 square yards.
Part IV: The Economics of an Estate
7.
a) Start with P = 2L + 2W. We must isolate L.
P - 2W = 2L
L = (P - 2W) / 2 or L = P/2 - W.
b) Using the rearranged formula:
L = (320 - 2(60)) / 2
L = (320 - 120) / 2
L = 200 / 2 = 100 yards.
8.
a) The fixed cost plus the per-guest cost must be less than or equal to the budget.
10 + 2g ≤ 40
b) Solve for g:
2g ≤ 40 - 10
2g ≤ 30
g ≤ 15
The maximum number of guests she can invite is 15.