Ask question

Ask question

All categories

2 years ago

11

if 1/4 of a gallon of paint is needed to paint 2/3 of a fence, how many gallons are needed to paint the entire fence? site:socra

tic.org

2 answers:

FinnZ [79.3K]2 years ago

8 0

Answer:3/8 gallons of paint is needed to paint the entire fence.

Step-by-step explanation:

Let x represent the entire area of the fence that is to be painted.

If 1/4 of a gallon of paint is needed to paint 2/3 of a fence, it means that 1/4 of a gallon of paint is needed to paint 2x/3

The number of gallons needed to paint the entire fence would be

(x/4)/(2x/3) = x/4 × 3/2x = 3/8 gallons

trasher [3.6K]2 years ago

7 0

$\bf \begin{array}{ccll} \stackrel{paint}{gallons}&fence\\ \cline{1-2} \\\frac{1}{4}&\frac{2}{3}\\\\ x&1 \end{array}\implies \cfrac{~~\frac{1}{4}~~}{x}=\cfrac{~~\frac{2}{3}~~}{1}\implies \cfrac{~~\frac{1}{4}~~}{\frac{x}{1}}=\cfrac{2}{3}\implies \cfrac{1}{4}\cdot \cfrac{1}{x}=\cfrac{2}{3} \\\\\\ \cfrac{1}{4x}=\cfrac{2}{3}\implies 3=8x\implies \cfrac{3}{8}=x$

You might be interested in

A light bulb is designed by revolving the graph of:

nadya68 [22]

Answer:

$\displaystyle 0.251327 \ in. \ of \ glass$

General Formulas and Concepts:

<u>Pre-Algebra</u>

Order of Operations: BPEMDAS

Brackets
Parenthesis
Exponents
Multiplication
Division
Addition
Subtraction

Left to Right

<u>Algebra I</u>

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

<u>Calculus</u>

Differentiation

Derivative Notation

Basic Power Rule:

f(x) = cxⁿ
f’(x) = c·nxⁿ⁻¹

Integration

Integration Property: $\displaystyle \int\limits^a_b {cf(x)} \, dx = c \int\limits^a_b {f(x)} \, dx$

Fundamental Theorem of Calculus: $\displaystyle \int\limits^a_b {f(x)} \, dx = F(b) - F(a)$

Area between Two Curves

Volumes of Revolution

Arc Length Formula: $\displaystyle AL = \int\limits^a_b {\sqrt{1+ [f'(x)]^2}} \, dx$

Surface Area Formula: $\displaystyle SA = 2\pi \int\limits^a_b {f(x) \sqrt{1+ [f'(x)]^2}} \, dx$

Step-by-step explanation:

<u>Step 1: Define</u>

$\displaystyle y = \frac{1}{3}x^{\frac{1}{2}} - x^{\frac{3}{2}}\\Interval: [0, \frac{1}{3}]$

<u>Step 2: Differentiate</u>

Basic Power Rule: $\displaystyle y' = \frac{1}{2} \cdot \frac{1}{3}x^{\frac{1}{2} - 1} - \frac{3}{2} \cdot x^{\frac{3}{2} - 1}$
[Derivative] Simplify: $\displaystyle y' = \frac{1}{6}x^{\frac{-1}{2}} - \frac{3}{2}x^{\frac{1}{2}}$
[Derivative] Simplify: $\displaystyle y' = \frac{1}{6\sqrt{x}} - \frac{3\sqrt{x}}{2}}$

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

Substitute [Surface Area]: $\displaystyle SA = 2\pi \int\limits^{\frac{1}{3}}_0 {(\frac{1}{3}x^{\frac{1}{2}} - x^{\frac{3}{2}}) \sqrt{1+ [\frac{1}{6\sqrt{x}} - \frac{3\sqrt{x}}{2}}]^2}} \, dx$
[Integral - √Radical] Expand/Add: $\displaystyle SA = 2\pi \int\limits^{\frac{1}{3}}_0 {(\frac{1}{3}x^{\frac{1}{2}} - x^{\frac{3}{2}}) \sqrt{\frac{81x^2+18x+1}{36x}} \, dx$
[Integral - √Radical] Factor: $\displaystyle SA = 2\pi \int\limits^{\frac{1}{3}}_0 {(\frac{1}{3}x^{\frac{1}{2}} - x^{\frac{3}{2}}) \sqrt{\frac{(9x + 1)^2}{36x}} \, dx$
[Integral - Simplify]: $\displaystyle SA = 2\pi \int\limits^{\frac{1}{3}}_0 {-\frac{|9x + 1|(3x - 1)}{18}} \, dx$
[Integral] Integration Property: $\displaystyle SA = \frac{- \pi}{9} \int\limits^{\frac{1}{3}}_0 {|9x + 1|(3x - 1)} \, dx$

<u>Step 4: Integrate Pt. 2</u>

[Integral] Define: $\displaystyle \int {|9x + 1|(3x - 1)} \, dx$
[Integral] Assumption of Positive/Correction Factors: $\displaystyle \frac{9x + 1}{|9x + 1|} \int {(9x + 1)(3x - 1)} \, dx$
[Integral] Expand - FOIL: $\displaystyle \frac{9x + 1}{|9x + 1|} \int {27x^2 - 6x - 1} \, dx$
[Integral] Integrate - Basic Power Rule: $\displaystyle \frac{9x + 1}{|9x + 1|} (9x^3 - 3x^2 - x)$
[Expression] Multiply: $\displaystyle \frac{(9x + 1)(9x^3 - 3x^2 - x)}{|9x + 1|}$

<u>Step 5: Integrate Pt. 3</u>

[Integral] Substitute/Integral - FTC: $\displaystyle SA = \frac{- \pi}{9} (\frac{(9x + 1)(9x^3 - 3x^2 - x)}{|9x + 1|})|\limits_{0}^{\frac{1}{3}}$
[Integrate] Evaluate FTC: $\displaystyle SA = \frac{- \pi}{9} (\frac{-1}{3})$
[Expression] Multiply: $\displaystyle SA = \frac{\pi}{27} \ ft^2$

<em>It is in ft² because it is given that our axis are in ft.</em>

<u>Step 6: Find Amount of Glass</u>

<em>Convert ft² to in² and multiply by 0.015 in (given) to find amount of glass.</em>

Convert ft² to in²: $\displaystyle \frac{\pi}{27} \ ft^2 \ \div 144 \ in^2/ft^2 = \frac{16 \pi}{3} \ in^2$
Multiply: $\displaystyle \frac{16 \pi}{3} \ in^2 \cdot 0.015 \ in = 0.251327 \ in. \ of \ glass$

And we have our final answer! Hope this helped on your Calc BC journey!

5 0

2 years ago

Assume that lines that appear to be tangent are tangent. O is the center of the circle. In degrees, what is the value of x?

Nastasia [14]

9514 1404 393

Answer:

145

Step-by-step explanation:

In this geometry, the central angle (x°) and the external angle (35°) are supplementary.

x° = 180° -35°

x = 145

4 0

2 years ago

Need some help here please...

vichka [17]

Answer:

The answer is (3,2) or x=3 and y=2

Step-by-step explanation:

5 0

2 years ago

Read 2 more answers

Please help, will give 5 stars

Alexus [3.1K]

Function 1
rate of change = (80-100)/(2-1) = -20
y = -20x + 120

function 2
y = -18x + 120
rate of change = -18

answer
function 2 shows a greater rate of change

7 0

2 years ago

Help please? 20 points

drek231 [11]

Answer:

125.6637061 or 125.7

Step-by-step explanation:

V=πr²h

Big: π(3)²(5) = 141.3716694

Hole: π(1)²(5) = 15.70796327

141.3716694

- <u>15.70796327</u>

125.6637061

7 0

2 years ago