Resources

Resources

Bytelearn - cat image with glasses

AI tutor

Welcome to Bytelearn!

Let’s check out your problem:

The area of a rectangular piece of metal is $36\,\text{square centimeters}$ . The perimeter is $26\,\text{centimeters}$ . What are the dimensions of the piece? $\newline$ $\_\_\_\_$ centimeters by $\_\_\_\_$ centimeters

View step-by-step help

View step-by-step help

Area and perimeter: word problems

Full solution

Q. The area of a rectangular piece of metal is

36\,\text{square centimeters}

. The perimeter is

26\,\text{centimeters}

. What are the dimensions of the piece?

\newline

\_\_\_\_

centimeters by

\_\_\_\_

centimeters

Define Variables: Let's denote the length of the rectangle as $L$ cm and the width as $W$ cm. We are given two equations based on the area and perimeter of the rectangle: $\newline$ $1$ . Area $(A) = L \times W$ $\newline$ $2$ . Perimeter $(P) = 2 \times (L + W)$ $\newline$ From the problem, we know: $\newline$ $A = 36 \, \text{cm}^2$ $\newline$ $P = 26 \, \text{cm}$ $\newline$ We can use these two equations to find the values of $L$ and $W$ .
Write Equations: First, let's write down the equations with the given values: $\newline$ $1$ . $L \times W = 36$ $\newline$ $2$ . $2 \times (L + W) = 26$ $\newline$ Now, we can simplify the second equation to find an expression for $L + W$ : $\newline$ $2 \times (L + W) = 26$ $\newline$ $L + W = \frac{26}{2}$ $\newline$ $L + W = 13$
Simplify Equations: We have two equations now: $\newline$ $1$ . $L \times W = 36$ $\newline$ $2$ . $L + W = 13$ $\newline$ We can use substitution or elimination to solve these equations. Let's express $W$ in terms of $L$ using the second equation: $\newline$ $W = 13 - L$
Substitute and Solve: Now, we substitute $W = 13 - L$ into the first equation: $\newline$ $L \times (13 - L) = 36$ $\newline$ Expanding this, we get a quadratic equation: $\newline$ $L^2 - 13L + 36 = 0$ $\newline$ To find the values of $L$ , we need to factor this quadratic equation.
Factor Quadratic Equation: Factoring the quadratic equation $L^2 - 13L + 36 = 0$ , we look for two numbers that multiply to $36$ and add up to $13$ . These numbers are $9$ and $4$ . $\newline$ So, the factors are: $\newline$ $(L - 9)(L - 4) = 0$ $\newline$ Setting each factor equal to zero gives us the possible values for $L$ : $\newline$ $L - 9 = 0$ or $L - 4 = 0$ $\newline$ $L = 9$ or $36$ $0$
Find Possible Values: Since $L$ can be either $9$ or $4$ , we need to determine the corresponding width for each case. If $L = 9$ cm, then $W = 13 - L = 13 - 9 = 4$ cm. If $L = 4$ cm, then $W = 13 - L = 13 - 4 = 9$ cm. $\newline$ Therefore, the dimensions of the rectangle can be either $9$ cm by $4$ cm or $4$ cm by $9$ cm. Both sets of dimensions satisfy the area and perimeter conditions given in the problem.

More problems from Area and perimeter: word problems

Question

The work done by stretching a certain spring increases by

0.13 x

joules per centimeter (where

x

is the displacement, in centimeters, beyond the spring's natural length).

\newline

How much work (in Joules) must be done in order to stretch the spring from

x=4

x=10

\newline

1

\newline

5

46

\newline

7

54

\newline

\mathbf{1 0 . 9 2}

\newline

\mathbf{1 5 . 0 8}

Posted 3 months ago

Question

Ahmed is modeling the area covered by a moss using the following equation:

\newline

A=t^{2}+5.8 t+9.41

\newline

A

is the area covered in square centimeters, negative values of

t

represent a number of days before noon on July

1

2014

, and positive values of

t

represent a number of days after noon on July

1

2014

. Which of the following equivalent expressions contains the number of days before noon on July

1

2014

, as a constant or coefficient, when the moss started with its smallest area of

1

square centimeter?

\newline

1

\newline

(t-(-2.9))^{2}+1

\newline

(t-(-1))^{2}+3.8 t+8.41

\newline

(t-(-3.9))(t-(-1.9))+2

\newline

t(t-(-5.8))+9.41

Posted 3 months ago

Question

3

03

Quiz: Volumes of Cylind

\newline

\newline

What is the volume of a cylinder with a radius of

3

feet and a height of

4

\newline

3.14

\newline

Enter your answer in the box

Posted 2 months ago

Question

Critical Thinking

\newline

& Problem Solving

\newline

Toy rubber balls are packaged in a cylinder that holds

3

balls. The diameter of each ball is

\newline

5.3\,\text{cm}

. Find the volume of the cylinder. Use

3.14

\newline

\pi

\newline

The volume of the cylinder is about

\newline

\square\,\text{cm}^3

\newline

(Round to the nearest tenth as needed.)

Posted 2 months ago

Question

The following information is available for them: Accounte Recelvable on December

31

(prior year) is

\$350,000

, Sales from Quarter

1

\$300,000

, Sales from Quarter

2

\$250,000

, Soles from Quarter

3

\$500,000

, and Sales from quarter

4

\$600,000

. Ueing budget standarde, the cash to be collected for the first quarter will consist of the accounte receivable balance on December

31

(prior year) and

70

\% of the sales made during the

1

st quarter. For future quartere,

70

\% of eales will be collected in each current quarter and

30

\% will be collected in the quarter after the cale.

\newline

How much cash will Buffalo Corporation collect next year?

\newline

1,930,000

\newline

2,000,000

\newline

1,470,000

\newline

1,820,000

Posted 2 months ago

Question

\newline

Viewing Saved Work Revert to Last Response

\newline

\frac{1}{3} \text{ Points}

\newline

\newline

PREVIOUS ANSWERS

\newline

8

7

035

\newline

Write and evaluate the definite integral that represents the area of the surface generated by revolving the curve on the indicated interval about the

\newline

\newline

\left\{\begin{array}{l}y=\frac{x^{3}}{18},\quad 2 \leq x \leq 5,\2\pi\int_{2}^{5}(\square)dx=\square\end{array}\right.

\newline

\newline

-\frac{1}{1} \text{ Points}

\newline

\newline

8

7

031

\newline

A gasoline tank is an oblate spheroid generated by revolving the region bounded by the graph of

Posted 2 months ago

Question

The radius of a quarter circle is

13

miles. What is the quarter circle's area?

\newline

\pi \approx 3.14

and round your answer to the nearest hundredth.

\newline

square

miles

\newline

\newline

\newline

Not feeling ready yet? These can help:

\newline

Area of circles

(103)

Posted 2 months ago

Question

9

8

\newline

\newline

\newline

9

8

10

Check Your Understanding

\newline

A cone has a slant height of

8\,\text{in}

and the radius of the base is

3\,\text{in}

. What is the lateral area, in square inches? Use

\pi=3.14

, Round your answer to the nearest hundredth.

Posted 2 months ago

Question

A cone has a slant height of

8\,\text{in}

and the radius of the base is

3\,\text{in}

. What is the lateral area, in square inches? Use

\pi=3.14

. Round your answer to the nearest hundredth.

Posted 2 months ago

Question

F

is shown with a diameter of

15

( ext{cm})

. Which is the area of circle

F

\pi=3.14

Posted 2 months ago