Question
upstudy study bank question image url

A rectangular box has dimensions 4 in by 4 in by 3 in. Increasing each dimension of the box by the same amount yields a new box with volume six times the old. Use the ALEKS graphing calculator to find how much each dimension of the original box was increased to create the new box. Round your answer to two decimal places.

Ask by Barker Fernandez. in the United States
Mar 15,2025

Upstudy AI Solution

Tutor-Verified Answer

Answer

Each dimension of the original box was increased by approximately 2.96 inches.

Solution

Let \( x \) be the amount (in inches) by which each dimension is increased. The new dimensions become: - Length: \( 4+x \) - Width: \( 4+x \) - Height: \( 3+x \) The original volume is \[ 48 = 4 \times 4 \times 3. \] Increasing the volume by six times leads to a new volume of \[ 6 \times 48 = 288. \] Thus, the equation for the new volume is: \[ (4+x)^2 (3+x) = 288. \] **Step 1. Expand the Left-hand Side** First, expand \((4+x)^2\): \[ (4+x)^2 = 16 + 8x + x^2. \] Multiply by \((3+x)\): \[ (16 + 8x + x^2)(3+x) = 16(3+x) + 8x(3+x) + x^2(3+x). \] Compute each term: \[ 16(3+x) = 48 + 16x, \] \[ 8x(3+x) = 24x + 8x^2, \] \[ x^2(3+x) = 3x^2 + x^3. \] Adding these: \[ x^3 + (8x^2 + 3x^2) + (16x + 24x) + 48 = x^3 + 11x^2 + 40x + 48. \] Thus, we have: \[ x^3 + 11x^2 + 40x + 48 = 288. \] **Step 2. Form the Cubic Equation** Subtract 288 from both sides: \[ x^3 + 11x^2 + 40x + 48 - 288 = 0, \] which simplifies to: \[ x^3 + 11x^2 + 40x - 240 = 0. \] **Step 3. Solve for \( x \)** We now need the positive real solution of: \[ x^3 + 11x^2 + 40x - 240 = 0. \] Using numerical methods (or a graphing calculator such as the ALEKS graphing calculator), we find that the positive solution is approximately: \[ x \approx 2.96. \] **Answer** The dimensions of the original box were increased by approximately: \[ \boxed{2.96 \text{ inches}}. \]

Answered by UpStudy AI and reviewed by a Professional Tutor

error msg
Explain
Simplify this solution

Mind Expander

To start, the original volume of the box is calculated as follows: \(V = length \times width \times height = 4 \, \text{in} \times 4 \, \text{in} \times 3 \, \text{in} = 48 \, \text{in}^3\). When each dimension is increased by an amount \(x\), the new dimensions become \(4 + x\), \(4 + x\), and \(3 + x\). The volume of the new box then equals \((4 + x)(4 + x)(3 + x)\). Setting the new volume equal to six times the original volume gives us the equation: \((4 + x)(4 + x)(3 + x) = 6 \times 48\). Simplifying that leads to \((4 + x)^2(3 + x) = 288\). From this point, using the ALEKS graphing calculator allows us to explore the solutions for \(x\). More hands-on, you can rearrange the equation, finding \(x\) where necessary. Ultimately, the value for \(x\) should be roughly around 2.00 inches, when calculated accurately. Considering how dimensions work, if we increase the box dimensions uniformly, it's a neat trick to remember how volume geometrically expands— increasing a single linear dimension has all the exponential effects in spite of it being just a few additional inches. Quick math reminders, such as the properties of cubes and boxes, can be incredibly useful when juggling larger problems! If you're diving into more advanced math or geometry, there are also many resources available that enhance your understanding of volume modifications. One excellent read is “The Joy of Geometry” by Alfred Posamentier, which offers engaging explorations of shapes and their properties while keeping the math entertaining. Alternatively, for pragmatic applications, consider studying box designs in various industries—like packaging or construction—to see how volume impacts real-world products and building methodology!

Try Premium now!
Try Premium and ask Thoth AI unlimited math questions now!
Maybe later Go Premium
Study can be a real struggle
Why not UpStudy it?
Select your plan below
Premium

You can enjoy

Start now
  • Step-by-step explanations
  • 24/7 expert live tutors
  • Unlimited number of questions
  • No interruptions
  • Full access to Answer and Solution
  • Full Access to PDF Chat, UpStudy Chat, Browsing Chat
Basic

Totally free but limited

  • Limited Solution
Welcome to UpStudy!
Please sign in to continue the Thoth AI Chat journey
Continue with Email
Or continue with
By clicking “Sign in”, you agree to our Terms of Use & Privacy Policy