Mathematical Methods · Unit 1 · Surds and quadratic functions · Quadratic functions
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A ball is thrown vertically upward from ground level. The height of the ball above ground, h(t) (in metres), at time t (in seconds) is modelled by the function h(t) = 24t - 4.9t². Which of the following statements correctly describes the domain and the maximum height of the ball?
Solve for \(x\) in the following equation. \(\log_2(x + 5) + \log_2(x - 1) = \log_2(3x + 7)\)
A rectangular garden bed is to be constructed against an existing straight wall. Fencing of total length 36 metres is available to enclose the other three sides of the garden bed. (a) Let \(x\) metres represent the width of the garden bed (perpendicular to the wall). Show that the area \(A\) square metres of the garden bed can be expressed as \(A = 36x - 2x^2\). (b) Determine the dimensions of the garden bed that will produce the maximum area. (c) Calculate the maximum area of the garden bed.
A landscape gardener is designing a rectangular garden bed with a perimeter of 28 metres. The area of the garden bed is given by the function $A(x) = x(14 - x)$ square metres, where $x$ is the length of one side in metres. (a) Expand and simplify the area function to express it in the form $A(x) = ax^2 + bx + c$. [1 mark] (b) Determine the value of $x$ that maximises the area of the garden bed. [1 mark] (c) Calculate the maximum area of the garden bed. [1 mark]