MaxWong

\(y = \dfrac{5(x + 4)(x + 5)}{(x - 2)(x - 5)}\)

Using the section formula directly,

The required point is \(\left(\dfrac{5(1) + 1(10)}{5 + 1}, \dfrac{5(-5) + 1(1)}{5 + 1}\right) = \left(\dfrac{5}{2}, -4\right)\)

The x-intercepts are at (1, 0) and (5, 0). This means the roots of the quadratic function are 1 and 5.

We can be sure that the quadratic function is of the form \(f(x) = C (x - 1)(x - 5)\) where C is a constant.

We can substitute the point (x, f(x)) = (0, 7) into the equation to find C.

7 = C (-1)(-5)

C = 7/5.

Therefore \(f(x) = \dfrac 75 (x - 1)(x - 5)\)

If you cut it like this, you get 16 pieces.

Answer

= (1 + 2 + 3 + ... + 99) - (2 + 4 + 6 + 8 + 10 + ... + 98) - (5 + 10 + 15 + 20 + ... + 95) + (10 + 20 + ... + 90)

= 4950 - 2450 - 950 + 450

= 2000

Oh yes. you're right. I was careless.

Total weight = 60 pounds.

Let W pounds be the weight of black licorice bits in the mixture.

Then the weight of orange gumdrops will be (60 - W) pounds.

Considering the price,

1.9W + 2.2(60 - W) = 2 * 60

132 - 0.3W = 120

0.3W = 12

W = 12/0.3 = 40

40 pounds of black licorice bits and 20 pounds of orange gumdrops should be used.

h = -16t^2 + 90t = -16(t^2 - 45/8 t) = -16((t - 45/16)^2 - (45/16)^2) = 2025/16 - 16 (t - 45/16)^2

The maximum height of the ball is 2025/16.

(The unit is missing from the question.)

Let L ft be the length of the shorter leg.

Length of longer leg = L + 9

Length of hypotenuse = (L + 9) + 9 = L + 18

By Pythagorean theorem,

L^2 + (L + 9)^2 = (L + 18)^2

2L^2 + 18L + 81 = L^2 + 36L + 324

L^2 - 18L - 243 = 0

(L - 27)(L + 9) = 0

L = 27 or L = -9 (rejected)

Therefore the shorter leg is 27 ft, longer leg is (27 + 9) ft = 36 ft, hypotenuse = (27 + 18) ft = 45 ft.

Observe which part of the graph is below the x-axis.

Answer: -2 ≤ x ≤ 3