Question

Question 1 and 2 please :) (exact form)

Answer 1

1. x = 6 cm

2. The coordinates of the x-intercept are (-15.798879, 0) and (8.177004, 0)

b) The maximum value of is y is approximately 459.874

The value of 'x' for which the maximum occurs is x is -3.8109375

c) 'k' is approximately -15.63748 or 'k' is approximately 8.0156

Explanation:

1. The area of rectangle ABCD - Area of triangle ABP = 114 cm²

Therefore, we have;

The area of rectangle ABCD = (2·x - 3)·(3·x + 1)

∴ The area of rectangle ABCD = 6·x² + 2·x - 9·x - 3 = 6·x² - 7·x - 3

The area of triangle ABP = 1/2 × (3·x + 1)·x = 3·x²/2 + x/2

The area of rectangle ABCD - Area of triangle ABP = 6·x² - 7·x - 3 - 3·x²/2 - x/2

6·x² - 7·x - 3 - 3·x²/2 + x/2 = 9·x²/2 - 13·x/2 - 3 = 114

9·x²/2 - 15·x/2 - 3 = 114

9·x² - 15·x - 6 - 228 = 0

9·x² - 15·x - 234 = 0

x = (15 ± √((-15)² - 4 × 9 × (-234)))/(2 × 9) = (15 ± 93)/18

∴ x = 6 or x = -13/3

Given that x is a natural number, we have that x = 6 cm

2. The given equation is 'y = -3.2·x² - 24.39·x + 413.4'

Therefore, we have;

The x-intercept are the values of 'x', where the value of 'y' is 0

Therefore, we have, at the x-intercept;

y = 0 = -3.2·x² - 24.39·x + 413.4

By the quadratic formula, we have;

x = (24.39 ± (√((-24.39)² - 4×(-3.2)×413.4)))/(2 × (-3.2))

x ≈ -15.798879 or x ≈ 8.177004

The coordinates of the x-intercept are;

(-15.798879, 0) and (8.177004, 0)

b) The maximum is given at dy/dx = 0

Therefore, at the maximum point, we have;

dy/dx = d(-3.2·x² - 24.39·x + 413.4)/dx = -6.4·x - 24.39 = 0

dy/dx = -6.4·x - 24.39 = 0

∴ x = 24.39/(-6.4) = -3.8109375

x = -3.8109375 at the maximum point

At the maximum point, we have;

y = -3.2·x² - 24.39·x + 413.4 = -3.2·(-3.8109375)² - 24.39·(-3.8109375) + 413.4 ≈ 459.874

y ≈ 459.874

Therefore, the maximum value of 'y' is given as follows

`y_(max)` ≈ 459.874

The value of 'x' for which the maximum occurs is x = -3.8109375

c) The coordinate of the point is (12.3, k), therefore, we have;

y = -3.2·x² - 24.39·x + 413.4

When y = 12.3, therefore, -3.2·x² - 24.39·x + 413.4 = 12.3

-3.2·x² - 24.39·x + 413.4 = 12.3

-3.2·x² - 24.39·x + 413.4 - 12.3 = -3.2·x² - 24.39·x + 401.1 = 0

By the quadratic formula, we have;

x = (24.39 ± √((-24.39)² - 4×(-3.2)×401.1))/(2 × (-3.2))

x ≈ -15.63748 or x ≈ 8.015600

Therefore, k ≈ -15.63748 or k ≈ 8.015600