Unit 7
11 marks · HardA circular garden has a radius of metres. A path of uniform width metres is built all around the edge of the garden, forming a larger circle.
The combined area of the garden and the path is . The area of just the garden is .
Write down an expression for the area of the ring-shaped path in terms of and , and hence form a quadratic equation in .
Find the value of , giving your answer to 3 significant figures.
Using your value for , solve the quadratic equation you formed in part (a) to find the width of the path. Give your answer to 2 decimal places.
Calculate the perimeter of the outer edge of the path. Give your answer to 1 decimal place and state the unit.
Work this out on paper, then compare with the answer and worked solution below. With a free account you can submit written answers for marking.
Show answer
(a) Total area = , Garden area = , area of path = . Quadratic: . So: → · must show Correct expression for area of ring, Substitution of values, Quadratic equation layout with x and r
(b) r = 9.00 m · must show Substitution into , Value correct to 3 sig figs
(c) x = 2.12 m · must show Quadratic solution for x, Correct substitution for r
(d) 69.7 m · must show Perimeter = , Value correct to 1 decimal place
Common mistakes: Incorrect quadratic setup (subtracting instead of equating total areas); Arithmetic or algebraic error when substituting for r; Forgetting to add r and x for the outer radius
Show mark scheme and worked solution
Marking points
M1 (a) · 1 mark · States or uses area of a circle correctly.
M2 (a) · 1 mark · Forms correct expressions for the larger and smaller circle areas.
M3 (a) · 1 mark · Correctly forms an equation for the area of the path.
A1 (a) · 1 mark · Reaches or equivalent.
M4 (b) · 1 mark · Sets up and attempts to solve for r.
A2 (b) · 1 mark · Finds to 3 significant figures.
M5 (c) · 1 mark · Correct substitution of r and sets up quadratic in x.
M6 (c) · 1 mark · Attempts to solve quadratic equation for x (using correct method: factorising, formula, or otherwise).
A3 (c) · 1 mark · Obtains x = 2.12 (to 2 decimal places)
M7 (d) · 1 mark · Uses correct formula for circumference: with their values. (follow-through)
A4 (d) · 1 mark · Final value for perimeter to 1 decimal place with unit. (follow-through)
Worked solution
a) Area of whole = . Area of garden = .
Area of path only = total area - garden area = 380 - 254 = 126
So
b) (to 3sf)
c) Substitute :
Solve quadratic (using formula):
Negative value not possible; so (2 dp)
d) The outer edge has radius m
So to 1 decimal place: 69.7 m