Question
upstudy study bank question image url

A food company originally sells cereal in boxes with dimensions 11 in by 7 in by 2.5 in. To make more profit, the company decreases each dimension of the box by \( x \) inches but keeps the price the same. If the new volume is 90 in \( ^{3} \) by how much was each dimension decreased?

Ask by Bryant Cox. in the United States
Mar 16,2025

Upstudy AI Solution

Tutor-Verified Answer

Answer

Each dimension was decreased by 1 inch.

Solution

Sign in to Unlock Answers for Free!

A Learning Platform Trusted by Millions of Real Students and Teachers.

star-icon Unlock

Answered by UpStudy AI and reviewed by a Professional Tutor

Mind Expander

To find the value of \( x \) by which each dimension was decreased, we first need to write the expression for the volume of the new box. The original volume \( V \) of the cereal box is calculated by multiplying the dimensions: \[ V = 11 \text{ in} \times 7 \text{ in} \times 2.5 \text{ in} = 192.5 \text{ in}^3 \] When the company decreases each dimension by \( x \), the new dimensions become \( (11 - x) \), \( (7 - x) \), and \( (2.5 - x) \). The volume of the new box can be expressed as: \[ V_{new} = (11 - x)(7 - x)(2.5 - x) \] We know the new volume is 90 in³, so we set up the equation: \[ (11 - x)(7 - x)(2.5 - x) = 90 \] To solve this, we can expand the left side: 1. First, multiply \( (11 - x)(7 - x) \): \[ (11 - x)(7 - x) = 77 - 11x - 7x + x^2 = x^2 - 18x + 77 \] 2. Now, multiply the result by \( (2.5 - x) \): \[ (x^2 - 18x + 77)(2.5 - x) = 2.5x^2 - 18x(2.5 - x) + 77(2.5 - x) \] This expands to: \[ 2.5x^2 - 45x + 18x^2 + 192.5 - 77x = 20.5x^2 - 122x + 192.5 \] Setting this equal to 90 gives: \[ 20.5x^2 - 122x + 192.5 = 90 \] Simplifying: \[ 20.5x^2 - 122x + 102.5 = 0 \] Now, we can use the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) where \( a = 20.5 \), \( b = -122 \), and \( c = 102.5 \): Calculating the discriminant: \[ b^2 - 4ac = (-122)^2 - 4(20.5)(102.5) = 14884 - 8390 = 6494 \] Calculating \( x \): \[ x = \frac{122 \pm \sqrt{6494}}{41} \] Calculating \( \sqrt{6494} \approx 80.6 \): \[ x \approx \frac{122 \pm 80.6}{41} \] This gives us two possible solutions: 1. \( x = \frac{202.6}{41} \approx 4.93 \) 2. \( x = \frac{41.4}{41} \approx 1.01 \) Since we can't decrease dimensions beyond zero, we choose \( x \approx 1.01 \) inches. Thus, the company decreases each dimension by approximately \( 1 \) inch.

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