IDNLearn.com is the place where your questions are met with thoughtful and precise answers. Our community provides accurate and timely answers to help you understand and solve any issue.
Sagot :
To solve this problem, let's break it down step-by-step:
### Step 1: Understand the Relationship
Given that the mass ([tex]\( m \)[/tex]) of a cylinder varies jointly as the square of the radius ([tex]\( r \)[/tex]) and inversely as the square root of the height ([tex]\( h \)[/tex]), we have the mathematical relationship:
[tex]\[ m \propto \frac{r^2}{\sqrt{h}} \][/tex]
This relationship can be written as:
[tex]\[ m = k \frac{r^2}{\sqrt{h}} \][/tex]
where [tex]\( k \)[/tex] is a constant of proportionality.
### Step 2: Define Initial Values
Let's assume the initial radius and height of the cylinder are:
[tex]\[ r_{\text{initial}} = r \][/tex]
[tex]\[ h_{\text{initial}} = h \][/tex]
Using these initial values, the initial mass [tex]\( m_{\text{initial}} \)[/tex] can be defined as:
[tex]\[ m_{\text{initial}} = k \frac{r_{\text{initial}}^2}{\sqrt{h_{\text{initial}}}} \][/tex]
### Step 3: Apply the Changes
The radius is reduced by 25%, and the height is increased by 21%. Hence, the new radius and height will be:
[tex]\[ r_{\text{new}} = r_{\text{initial}} \times (1 - 0.25) = 0.75 \times r_{\text{initial}} \][/tex]
[tex]\[ h_{\text{new}} = h_{\text{initial}} \times (1 + 0.21) = 1.21 \times h_{\text{initial}} \][/tex]
### Step 4: Calculate the New Mass
Using the new values of radius and height, the new mass [tex]\( m_{\text{new}} \)[/tex] is:
[tex]\[ m_{\text{new}} = k \frac{r_{\text{new}}^2}{\sqrt{h_{\text{new}}}} \][/tex]
Substituting the new values:
[tex]\[ m_{\text{new}} = k \frac{(0.75r_{\text{initial}})^2}{\sqrt{1.21h_{\text{initial}}}} \][/tex]
[tex]\[ m_{\text{new}} = k \frac{0.5625 r_{\text{initial}}^2}{\sqrt{1.21} \sqrt{h_{\text{initial}}}} \][/tex]
[tex]\[ m_{\text{new}} = k \frac{0.5625 r_{\text{initial}}^2}{1.1 \sqrt{h_{\text{initial}}}} \][/tex]
### Step 5: Compare the Initial and New Mass
Let's express [tex]\( m_{\text{initial}} \)[/tex] and [tex]\( m_{\text{new}} \)[/tex] in terms of each other:
[tex]\[ m_{\text{initial}} = k \frac{r_{\text{initial}}^2}{\sqrt{h_{\text{initial}}}} \][/tex]
[tex]\[ m_{\text{new}} = k \frac{0.5625 r_{\text{initial}}^2}{1.1 \sqrt{h_{\text{initial}}}} \][/tex]
By simplifying the ratio of [tex]\( m_{\text{new}} \)[/tex] to [tex]\( m_{\text{initial}} \)[/tex]:
[tex]\[ \frac{m_{\text{new}}}{m_{\text{initial}}} = \frac{\frac{0.5625 r_{\text{initial}}^2}{1.1 \sqrt{h_{\text{initial}}}}}{\frac{r_{\text{initial}}^2}{\sqrt{h_{\text{initial}}}}} = \frac{0.5625}{1.1} \][/tex]
[tex]\[ \frac{m_{\text{new}}}{m_{\text{initial}}} = 0.5113636 \][/tex]
### Step 6: Calculate the Percentage Change in Mass
The percentage change in mass is given by:
[tex]\[ \text{Percentage change} = \left( \frac{m_{\text{new}} - m_{\text{initial}}}{m_{\text{initial}}} \right) \times 100 \][/tex]
[tex]\[ \text{Percentage change} = \left( \frac{0.5113636 m_{\text{initial}} - m_{\text{initial}}}{m_{\text{initial}}} \right) \times 100 \][/tex]
[tex]\[ \text{Percentage change} = \left( 0.5113636 - 1 \right) \times 100 \][/tex]
[tex]\[ \text{Percentage change} = -48.8636\% \][/tex]
### Final Answer
The percentage change in the mass of the cylinder is -48.86%. This means the mass of the cylinder is reduced by approximately 48.86% due to the changes in radius and height.
### Step 1: Understand the Relationship
Given that the mass ([tex]\( m \)[/tex]) of a cylinder varies jointly as the square of the radius ([tex]\( r \)[/tex]) and inversely as the square root of the height ([tex]\( h \)[/tex]), we have the mathematical relationship:
[tex]\[ m \propto \frac{r^2}{\sqrt{h}} \][/tex]
This relationship can be written as:
[tex]\[ m = k \frac{r^2}{\sqrt{h}} \][/tex]
where [tex]\( k \)[/tex] is a constant of proportionality.
### Step 2: Define Initial Values
Let's assume the initial radius and height of the cylinder are:
[tex]\[ r_{\text{initial}} = r \][/tex]
[tex]\[ h_{\text{initial}} = h \][/tex]
Using these initial values, the initial mass [tex]\( m_{\text{initial}} \)[/tex] can be defined as:
[tex]\[ m_{\text{initial}} = k \frac{r_{\text{initial}}^2}{\sqrt{h_{\text{initial}}}} \][/tex]
### Step 3: Apply the Changes
The radius is reduced by 25%, and the height is increased by 21%. Hence, the new radius and height will be:
[tex]\[ r_{\text{new}} = r_{\text{initial}} \times (1 - 0.25) = 0.75 \times r_{\text{initial}} \][/tex]
[tex]\[ h_{\text{new}} = h_{\text{initial}} \times (1 + 0.21) = 1.21 \times h_{\text{initial}} \][/tex]
### Step 4: Calculate the New Mass
Using the new values of radius and height, the new mass [tex]\( m_{\text{new}} \)[/tex] is:
[tex]\[ m_{\text{new}} = k \frac{r_{\text{new}}^2}{\sqrt{h_{\text{new}}}} \][/tex]
Substituting the new values:
[tex]\[ m_{\text{new}} = k \frac{(0.75r_{\text{initial}})^2}{\sqrt{1.21h_{\text{initial}}}} \][/tex]
[tex]\[ m_{\text{new}} = k \frac{0.5625 r_{\text{initial}}^2}{\sqrt{1.21} \sqrt{h_{\text{initial}}}} \][/tex]
[tex]\[ m_{\text{new}} = k \frac{0.5625 r_{\text{initial}}^2}{1.1 \sqrt{h_{\text{initial}}}} \][/tex]
### Step 5: Compare the Initial and New Mass
Let's express [tex]\( m_{\text{initial}} \)[/tex] and [tex]\( m_{\text{new}} \)[/tex] in terms of each other:
[tex]\[ m_{\text{initial}} = k \frac{r_{\text{initial}}^2}{\sqrt{h_{\text{initial}}}} \][/tex]
[tex]\[ m_{\text{new}} = k \frac{0.5625 r_{\text{initial}}^2}{1.1 \sqrt{h_{\text{initial}}}} \][/tex]
By simplifying the ratio of [tex]\( m_{\text{new}} \)[/tex] to [tex]\( m_{\text{initial}} \)[/tex]:
[tex]\[ \frac{m_{\text{new}}}{m_{\text{initial}}} = \frac{\frac{0.5625 r_{\text{initial}}^2}{1.1 \sqrt{h_{\text{initial}}}}}{\frac{r_{\text{initial}}^2}{\sqrt{h_{\text{initial}}}}} = \frac{0.5625}{1.1} \][/tex]
[tex]\[ \frac{m_{\text{new}}}{m_{\text{initial}}} = 0.5113636 \][/tex]
### Step 6: Calculate the Percentage Change in Mass
The percentage change in mass is given by:
[tex]\[ \text{Percentage change} = \left( \frac{m_{\text{new}} - m_{\text{initial}}}{m_{\text{initial}}} \right) \times 100 \][/tex]
[tex]\[ \text{Percentage change} = \left( \frac{0.5113636 m_{\text{initial}} - m_{\text{initial}}}{m_{\text{initial}}} \right) \times 100 \][/tex]
[tex]\[ \text{Percentage change} = \left( 0.5113636 - 1 \right) \times 100 \][/tex]
[tex]\[ \text{Percentage change} = -48.8636\% \][/tex]
### Final Answer
The percentage change in the mass of the cylinder is -48.86%. This means the mass of the cylinder is reduced by approximately 48.86% due to the changes in radius and height.
We value your presence here. Keep sharing knowledge and helping others find the answers they need. This community is the perfect place to learn together. IDNLearn.com has the answers you need. Thank you for visiting, and we look forward to helping you again soon.
