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2 years ago

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Ms. Walker's science class is doing an egg drop experiment from the balcony of their school. Each egg is protected by a contrapt

ion that the students collectively designed. The height of the egg, in feet, after x seconds is given by the expression below.
-16x^2+35

What do the zeros of the expression represent?

A.
the horizontal distance traveled by the egg
B.
the time at which the egg reaches its maximum height
C.
the maximum height of the egg
D.
the time at which the egg reaches the ground

1 answer:

Brrunno [24]2 years ago

5 0

Answer:

The answer is (D) ⇒ the time at which the egg reaches the ground

Step-by-step explanation:

∵ h(x) = -16x² + 35 ⇒ x is the time in seconds

∵ The zeroes of the expression means the value of x when h(x) = 0

and h is the height of the egg form the ground

∴ -16x² + 35 = 0

∴ 16x² = 35

∴ x² = 35/16 = 2.1875

∴ x = √2.1875 = 1.479

∴ The egg reaches the ground after 1.479 sec.

∴ The zeroes of the expression means the time at which the

egg reaches the ground

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General Formulas and Concepts:

<u>Alg I</u>

Terms/Coefficients
Factor
Exponential Rule [Dividing]: $\displaystyle \frac{b^m}{b^n} = b^{m - n}$

<u>Pre-Calc</u>

[Right Triangle Only] Pythagorean Theorem: a² + b² = c²

a is a leg
b is a leg
c is hypotenuse

Trigonometric Ratio: $\displaystyle sec(\theta) = \frac{1}{cos(\theta)}$

Trigonometric Identity: $\displaystyle tan^2\theta + 1 = sec^2\theta$

TI: $\displaystyle sin(2x) = 2sin(x)cos(x)$

TI: $\displaystyle cos^2(\theta) = \frac{cos(2x) + 1}{2}$

<u>Calc</u>

Integration Rule [Reverse Power Rule]: $\displaystyle \int {x^n} \, dx = \frac{x^{n + 1}}{n + 1} + C$

Integration Property [Multiplied Constant]: $\displaystyle \int {cf(x)} \, dx = c \int {f(x)} \, dx$

IP [Addition/Subtraction]: $\displaystyle \int {[f(x) \pm g(x)]} \, dx = \int {f(x)} \, dx \pm \int {g(x)} \, dx$

U-Substitution

U-Trig Substitution: x² + a² → x = atanθ

Step-by-step explanation:

<u>Step 1: Define</u>

$\displaystyle \int {\frac{dx}{(81x^2 + 4)^2}}$

<u>Step 2: Identify Sub Variables Pt.1</u>

Rewrite integral [factor expression]:

$\displaystyle \int {\frac{dx}{[(9x)^2 + 4]^2}}$

Identify u-trig sub:

$\displaystyle x = atan\theta\\9x = 2tan\theta \rightarrow x = \frac{2}{9}tan\theta\\dx = \frac{2}{9}sec^2\theta d\theta$

Later, back-sub θ (integrate w/ respect to <em>x</em>):

$\displaystyle tan\theta = \frac{9x}{2} \rightarrow \theta = arctan(\frac{9x}{2})$

<u>Step 3: Integrate Pt.1</u>

[Int] Sub u-trig variables: $\displaystyle \int {\frac{\frac{2}{9}sec^2\theta}{[(2tan\theta)^2 + 4]^2}} \ d\theta$
[Int] Rewrite [Int Prop - MC]: $\displaystyle \frac{2}{9} \int {\frac{sec^2\theta}{[(2tan\theta)^2 + 4]^2}} \ d\theta$
[Int] Evaluate exponents: $\displaystyle \frac{2}{9} \int {\frac{sec^2\theta}{[4tan^2\theta + 4]^2}} \ d\theta$
[Int] Factor: $\displaystyle \frac{2}{9} \int {\frac{sec^2\theta}{[4(tan^2\theta + 1)]^2}} \ d\theta$
[Int] Rewrite [TI]: $\displaystyle \frac{2}{9} \int {\frac{sec^2\theta}{[4sec^2\theta]^2}} \ d\theta$
[Int] Evaluate exponents: $\displaystyle \frac{2}{9} \int {\frac{sec^2\theta}{16sec^4\theta} \ d\theta$
[Int] Rewrite [Int Prop - MC]: $\displaystyle \frac{1}{72} \int {\frac{sec^2\theta}{sec^4\theta} \ d\theta$
[Int] Divide [ER - D]: $\displaystyle \frac{1}{72} \int {\frac{1}{sec^2\theta} \ d\theta$
[Int] Rewrite [TR]: $\displaystyle \frac{1}{72} \int {cos^2\theta} \ d\theta$
[Int] Rewrite [TI]: $\displaystyle \frac{1}{72} \int {\frac{cos(2\theta) + 1}{2}} \ d\theta$
[Int] Rewrite [Int Prop - MC]: $\displaystyle \frac{1}{144} \int {cos(2\theta) + 1} \ d\theta$
[Int] Rewrite [Int Prop - A/S]: $\displaystyle \frac{1}{144} [\int {cos(2\theta) \ d\theta + \int {1} \ d\theta]$

<u>Step 4: Identify Sub Variables Pt.2</u>

Determine u-sub for trig int:

u = 2θ

du = 2dθ

<u>Step 5: Integrate Pt.2</u>

[Ints] Rewrite [Int Prop - MC]: $\displaystyle \frac{1}{144} [\frac{1}{2} \int {2cos(2\theta) \ d\theta + \int {1 \theta ^0} \ d\theta]$
[Int] U-Sub: $\displaystyle \frac{1}{144} [\frac{1}{2} \int {cos(u) \ du + \int {1 \theta ^0} \ d\theta]$
[Ints] Integrate [Trig/Int Rule - RPR]: $\displaystyle \frac{1}{144} [\frac{1}{2} sin(u) + \theta + C]$
[Expression] Back Sub: $\displaystyle \frac{1}{144} [\frac{1}{2} sin(2 \theta) + arctan(\frac{9x}{2}) + C]$
[Exp] Rewrite [TI]: $\displaystyle \frac{1}{144} [\frac{1}{2}(2sin(\theta)cos(\theta)) + arctan(\frac{9x}{2}) + C]$
[Exp] Multiply: $\displaystyle \frac{1}{144} [sin(\theta)cos(\theta) + arctan(\frac{9x}{2}) + C]$
[Exp] Back Sub: $\displaystyle \frac{1}{144} [sin(arctan(\frac{9x}{2}))cos(arctan(\frac{9x}{2})) + arctan(\frac{9x}{2}) + C]$

<u>Step 6: Triangle</u>

Find trig values:

$\displaystyle tan\theta = \frac{9x}{2}$

$\displaystyle \theta = arctan(\frac{9x}{2})$

tanθ = opposite / adjacent; solve hypotenuse of right triangle, determine trig ratios:

sinθ = opposite / hypotenuse

cosθ = adjacent / hypotenuse

Leg <em>a</em> = 2

Leg <em>b</em> = 9x

Leg <em>c</em> = ?

Sub variables [PT]: $\displaystyle 2^2 + (9x)^2 = c^2$
Evaluate exponents: $\displaystyle 4 + 81x^2 = c^2$
[Equality Property] Square root both sides: $\displaystyle \sqrt{4 + 81x^2} = c$
Rewrite: $c = \sqrt{81x^2 + 4}$

Substitute into trig ratios:

$\displaystyle sin\theta = \frac{9x}{\sqrt{81x^2 + 4}}$

$\displaystyle cos\theta = \frac{2}{\sqrt{81x^2 + 4}}$

<u>Step 7: Integrate Pt.3</u>

[Exp] Sub variables [TR]: $\displaystyle \frac{1}{144} [\frac{9x}{\sqrt{81x^2 + 4}} \cdot \frac{2}{\sqrt{81x^2 + 4}} + arctan(\frac{9x}{2}) + C]$
[Exp] Multiply: $\displaystyle \frac{1}{144} [\frac{18x}{81x^2 + 4} + arctan(\frac{9x}{2}) + C]$
[Exp] Distribute: $\displaystyle \frac{1}{144}arctan(\frac{9x}{2}) + \frac{x}{8(81x^2 + 4)} + C$

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2 years ago