The Sears Tower, at 1,451 feet, is one of the tallest structures in the United States. A penny is thrown from the top of the tow

Explanation:

Step-by-step clarification:

Referring to step 6

(b² — 4ac) / 4a² = (x + b/2a)²

The mistake in the question is that it should be (x + b/2a)²

According to step 7

±√(b² —4ac) /2a = x + b/2a

The error in the question is that it should be divided by 2a, not 1a.

1. The transition from step 6 to step 7 involves taking the square roots of both sides

(b² — 4ac) / 4a² = (x + b/2a)²

Taking the square of both sides

√(b²—4ac) / √4a² = √(x + b/2a)²

√(b²—4ac) / 2a = x + b/2a

This forms step 7 correctly.

Next, subtracting b/2a from both sides

√(b²—4ac) / 2a - b/2a= x + b/2a -b/2a

√(b²—4ac) / 2a — b/2a = x

(√(b²—4ac)  — b)/2a = x

x = [—b ± √(b²—4ac)] / 2a

This gives the desired formula.

The discriminant is D = b²—4ac.

Response:

Refer to the figure attached.

Step-by-step explanation:

Each portion is treated as a function.

Let x represent the hours worked in a day

The company packs 80 boxes of toys each hour during the first three hours. Thus, y₁ = 80x, for x ∈ [0,3]

After three hours, the total boxes packed = 80 * 3 = 240

They pause packaging for two hours for a training session. Therefore, y₂ = 240, for x ∈ [3,5]

Next, for the subsequent four hours, they pack 20 boxes of toys hourly

y₃ = 20x + c, for x ∈ [5,9]

To determine c, equate y₂ and y₃ at x = 5

∴ 240 = 20 * 5 + c

∴ c = 240 - 20 * 5 = 240 - 100 = 140

∴ y₃ = 20x + 140, for x ∈ [5,9]

Consequently,

y₁ = 80x, for x ∈ [0,3]

y₂ = 240, for x ∈ [3,5]

y₃ = 20x + 140, for x ∈ [5,9]

The graph plotting the piecewise function that depicts the described situation is shown in the attached illustration.