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BabaBlast [244]

2 years ago

8

6)a garden was square, but last year she made it makes or shortening one side 5 meters and the other 8 meters. If the area of th

e smaller garden is 180 m², then what was the area of her square garden

1 answer:

Alekssandra [29.7K]2 years ago

5 0

Answer:

let dimention of original garden be x

then (x-8)(x-5) = 180

x^2 -13x -140 = 0

(x-20)(x+7)=0

x= 20 or x = -7

x = 20

original area = (20)^2

= 400 m^2

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Use the limit definition of the derivative to find the slope of the tangent line to the curve

ale4655 [162]

Answer:

$\displaystyle f'(4) = 63$

General Formulas and Concepts:

<u>Pre-Algebra</u>

Order of Operations: BPEMDAS

Brackets
Parenthesis
Exponents
Multiplication
Division
Addition
Subtraction

Left to Right<u> </u>

Distributive Property

<u>Algebra I</u>

Expand by FOIL (First Outside Inside Last)
Factoring
Function Notation
Terms/Coefficients

<u>Calculus</u>

Derivatives

The definition of a derivative is the slope of the tangent line.

Limit Definition of a Derivative: $\displaystyle f'(x)= \lim_{h \to 0} \frac{f(x+h)-f(x)}{h}$

Step-by-step explanation:

<u>Step 1: Define</u>

f(x) = 7x² + 7x + 3

Slope of tangent line at x = 4

<u>Step 2: Differentiate</u>

Substitute in function [Limit Definition of a Derivative]: $\displaystyle f'(x)= \lim_{h \to 0} \frac{[7(x + h)^2 + 7(x + h) + 3]-(7x^2 + 7x + 3)}{h}$
[Limit - Fraction] Expand [FOIL]: $\displaystyle f'(x)= \lim_{h \to 0} \frac{[7(x^2 + 2xh + h^2) + 7(x + h) + 3]-(7x^2 + 7x + 3)}{h}$
[Limit - Fraction] Distribute: $\displaystyle f'(x)= \lim_{h \to 0} \frac{[7x^2 + 14xh + 7h^2 + 7x + 7h + 3] - 7x^2 - 7x - 3}{h}$
[Limit - Fraction] Combine like terms (x²): $\displaystyle f'(x)= \lim_{h \to 0} \frac{14xh + 7h^2 + 7x + 7h + 3 - 7x - 3}{h}$
[Limit - Fraction] Combine like terms (x): $\displaystyle f'(x)= \lim_{h \to 0} \frac{14xh + 7h^2 + 7h + 3 - 3}{h}$
[Limit - Fraction] Combine like terms: $\displaystyle f'(x)= \lim_{h \to 0} \frac{14xh + 7h^2 + 7h}{h}$
[Limit - Fraction] Factor: $\displaystyle f'(x)= \lim_{h \to 0} \frac{h(14x + 7h + 7)}{h}$
[Limit - Fraction] Simplify: $\displaystyle f'(x)= \lim_{h \to 0} 14x + 7h + 7$
[Limit] Evaluate: $\displaystyle f'(x) = 14x + 7$

<u>Step 3: Find Slope</u>

Substitute in <em>x</em>: $\displaystyle f'(4) = 14(4) + 7$
Multiply: $\displaystyle f'(4) = 56 + 7$
Add: $\displaystyle f'(4) = 63$

This means that the slope of the tangent line at x = 4 is equal to 63.

Hope this helps!

Topic: Calculus AB/1

Unit: Chapter 2 - Definition of a Derivative

(College Calculus 10e)

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2 years ago

The initial population of a town is 4500, and it grows with a doubling time of 10 years. What will the population be in 12 years

Rufina [12.5K]

In the given problem, there are several vital informations that have been provided. based on these information's the answer to the asked question can be easily reached. It is given that the initial population of the town is 4500. It doubles in every 10 years. The number of population in 12 years is needed to be found.
Initial population of the town = 4500
In 10 years the population of the town = 4500 * 2
= 9000
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= 10800
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2 years ago