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In this question, we seek to determine the area of a rectangular domain, R, with base on the x-axis and inscribed between the curve C:y=2 x and the circle of radius 2 centred in the origin, S:x²+ y² =4, and solve an optimisation problem related to it. The domain lies in the first quadrant. Let a be the x-coordinate of the top-left vertex of R and b the x-coordinate of the top-right vertex. a) By considering a line parallel to the x-axis and the vertices of R, or otherwise, find an equation relating the square of a and the square of b. We have that b^2= b) State the width, w, and height, h, of R as a function of a. We find that w = and h = c) State the area, A of R as a function of a: A(a) = d) The rectangular domain represents a flat garden whose cost in £ is given by C= a² + 3 b² +8 a. Find the maximum cost of the garden. The maximum cost is C= £49✓ Enter your approximate numerical answer rounding to the nearest pence. Does the most expensive garden correspond to the largest one? Justify your answer

In this question, we seek to determine the area of a rectangular domain, R, with base on the x-axis and inscribed between the curve C:y=2 x and the circle of radius 2 centred in the origin, S:x²+ y² =4, and solve an optimisation problem related to it. The domain lies in the first quadrant. Let a be the x-coordinate of the top-left vertex of R and b the x-coordinate of the top-right vertex. a) By considering a line parallel to the x-axis and the vertices of R, or otherwise, find an equation relating the square of a and the square of b. We have that b^2= b) State the width, w, and height, h, of R as a function of a. We find that w = and h = c) State the area, A of R as a function of a: A(a) = d) The rectangular domain represents a flat garden whose cost in £ is given by C= a² + 3 b² +8 a. Find the maximum cost of the garden. The maximum cost is C= £49✓ Enter your approximate numerical answer rounding to the nearest pence. Does the most expensive garden correspond to the largest one? Justify your answer

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter11: Topics From Analytic Geometry
Section11.1: Parabolas
Problem 33E
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In this question, we seek to determine the area of a rectangular domain, R, with base on the x-axis and inscribed between the curve C:y=2 x and the circle of radius 2 centred in the origin, S:x² + y² =4, and solve an optimisation problem related to it. The domain lies in the first quadrant. Let a be the x-coordinate of the top-left vertex of R and b the x-coordinate of the top-right vertex. a) By considering a line parallel to the x-axis and the vertices of R, or otherwise, find an equation relating the square of a and the square of b. We have that b^2 = b) State the width, w, and height, h, of R as a function of a. We find that w = and h = c) State the area, A of R as a function of a: A(a) = d) The rectangular domain represents a flat garden whose cost in £ is given by C= a² + 3 b² + 8 a. Find the maximum cost of the garden. The maximum cost is C= £ 4.0 Enter your approximate numerical answer rounding to the nearest pence. Does the most expensive garden correspond to the largest one? Justify your answer
Transcribed Image Text:In this question, we seek to determine the area of a rectangular domain, R, with base on the x-axis and inscribed between the curve C:y=2 x and the circle of radius 2 centred in the origin, S:x² + y² =4, and solve an optimisation problem related to it. The domain lies in the first quadrant. Let a be the x-coordinate of the top-left vertex of R and b the x-coordinate of the top-right vertex. a) By considering a line parallel to the x-axis and the vertices of R, or otherwise, find an equation relating the square of a and the square of b. We have that b^2 = b) State the width, w, and height, h, of R as a function of a. We find that w = and h = c) State the area, A of R as a function of a: A(a) = d) The rectangular domain represents a flat garden whose cost in £ is given by C= a² + 3 b² + 8 a. Find the maximum cost of the garden. The maximum cost is C= £ 4.0 Enter your approximate numerical answer rounding to the nearest pence. Does the most expensive garden correspond to the largest one? Justify your answer
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