Question

review: writing and solving linear equations
multiple choice
identify the choice that best completes the statement or answers the question.

1. geometry the formula for the volume of a cone is ( v = \frac{1}{3}pi r^2 h ), where ( v ) represents the volume, ( r ) represents the radius of the base, and ( h ) represents the height. what is the height of a cone with a volume of 110 cubic centimeters and a base with a radius of 5 centimeters?

a.
b.
c.
d.

1. twice a number ( m ) minus three equals the sum of ( m ) and five.

a. ( 2(m - 3) = m + 5 )
b. ( 2m - 3 = m + 5 )
c. ( 2m - 3 = 5m )
d. ( 2(m - 3) = 5m )

Explanation:

Problem 2:

Step1: Analyze the problem statement

The problem says "Twice a number \( m \) minus three equals the sum of \( m \) and five". Let's break down each part:

• "Twice a number \( m \)" is \( 2m \).
• "minus three" gives \( 2m - 3 \).
• "equals the sum of \( m \) and five" means \( 2m - 3 = m + 5 \)? Wait, no, wait. Wait, the original wording: "Twice a number \( m \) minus three equals the sum of \( m \) and five". Wait, let's re - read: "Twice a number \( m \) minus three equals the sum of \( m \) and five". Wait, "Twice (a number \( m \) minus three)" would be \( 2(m - 3) \), but the wording is "Twice a number \( m \) minus three" which is \( 2m-3 \), and "the sum of \( m \) and five" is \( m + 5 \). Wait, but let's check the options. Option b is \( 2m - 3=m + 5 \), option a is \( 2(m - 3)=m + 5 \), option c is \( 2m-3 = 5m \), option d is \( 2(m - 3)=5m \).

Wait, the correct translation of "Twice a number \( m \) minus three equals the sum of \( m \) and five" is:

• "Twice a number \( m \)" is \( 2m \)
• "minus three" is \( 2m-3 \)
• "equals" is \( = \)
• "the sum of \( m \) and five" is \( m + 5 \)

So the equation is \( 2m-3=m + 5 \), which is option b. Wait, but let's check the wording again. Wait, maybe I misread. The problem says "Twice a number \( m \) minus three equals the sum of \( m \) and five". So:

Left - hand side: Twice \( m \) minus three \( = 2m-3 \)

Right - hand side: Sum of \( m \) and five \( = m + 5 \)

So the equation is \( 2m-3=m + 5 \), which is option b.

Step2: Verify the options

• Option a: \( 2(m - 3)=m + 5 \) translates to "Twice (a number \( m \) minus three) equals the sum of \( m \) and five", which is not the same as the given statement.
• Option b: \( 2m-3=m + 5 \) translates to "Twice a number \( m \) minus three equals the sum of \( m \) and five", which matches.
• Option c: \( 2m-3 = 5m \) translates to "Twice a number \( m \) minus three equals five times \( m \)", which is incorrect.
• Option d: \( 2(m - 3)=5m \) translates to "Twice (a number \( m \) minus three) equals five times \( m \)", which is incorrect.

Step1: Recall the volume formula of a cone

The volume formula of a cone is \( V=\frac{1}{3}\pi r^{2}h \), where \( V \) is the volume, \( r \) is the radius of the base, and \( h \) is the height. We are given that \( V = 110\pi \) cubic centimeters (wait, the original problem says \( V = 110 \) cubic centimeters? Wait, the user's image shows "110 cubic centimeters" and \( r = 5 \) centimeters. Let's substitute the values into the formula.

Given \( V=\frac{1}{3}\pi r^{2}h \), \( V = 110 \) (wait, maybe it's \( 110\pi \)? Let's check the formula. If \( V=\frac{1}{3}\pi r^{2}h \), and \( r = 5 \), then:

\( 110=\frac{1}{3}\pi(5)^{2}h \) (if \( V = 110 \)) or \( 110\pi=\frac{1}{3}\pi(5)^{2}h \) (if \( V = 110\pi \)). Let's assume \( V = 110\pi \) (maybe a typo in the image). Then:

Step2: Solve for \( h \)

Substitute \( V = 110\pi \) and \( r = 5 \) into \( V=\frac{1}{3}\pi r^{2}h \)

\( 110\pi=\frac{1}{3}\pi(5)^{2}h \)

First, divide both sides by \( \pi \):

\( 110=\frac{1}{3}(25)h \)

\( 110=\frac{25}{3}h \)

Multiply both sides by \( \frac{3}{25} \):

\( h=\frac{110\times3}{25}=\frac{330}{25}=\frac{66}{5} = 13.2 \)

But since the options are not given (the image cuts off the options for problem 1), we can only show the process.

If we assume \( V = 110 \) (without \( \pi \)):

\( 110=\frac{1}{3}\pi(25)h \)

\( h=\frac{110\times3}{25\pi}=\frac{66}{5\pi}\approx4.2 \)

But since the options are missing, we can't provide the final answer for problem 1. For problem 2, the answer is option b.

Answer:

b. \( 2m - 3=m + 5 \)

Problem 1: