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a caterpillar can crawl 1/2 inch in 1/4 of a minute which process should be used to determine the unit rate (distance traveled in one minute.)

a. 1/2-1/4
b. 1/2÷1/4
c. 1/4÷1/4
d. 1/2+1/4

9 answers

To determine the unit rate, which is the distance the caterpillar travels in one minute, you should use option b. \( \frac{1}{2} \div \frac{1}{4} \).

This is because you want to find out how much distance the caterpillar can travel in one full minute given that it crawls \( \frac{1}{2} \) inch in \( \frac{1}{4} \) minute. Dividing the distance by the time will give you the rate per minute.

So, you would set it up as: \[ \text{Unit Rate} = \frac{\text{Distance}}{\text{Time}} = \frac{\frac{1}{2}}{\frac{1}{4}} \]

Calculating this gives you: \[ \frac{1}{2} \div \frac{1}{4} = \frac{1}{2} \times \frac{4}{1} = 2 \]

Thus, the caterpillar crawls 2 inches in one minute.

rise from (-5,0) to (0,4) = __
run from (-5,0) to (0,4) = __
slope is equal to rise/run=

To calculate the rise, run, and slope between the points \((-5, 0)\) and \((0, 4)\), we need to follow these steps:

  1. Calculate the rise:

    • Rise is the change in the y-coordinates: \[ \text{Rise} = y_2 - y_1 = 4 - 0 = 4 \]

  2. Calculate the run:

    • Run is the change in the x-coordinates: \[ \text{Run} = x_2 - x_1 = 0 - (-5) = 0 + 5 = 5 \]

  3. Calculate the slope:

    • The slope (m) is defined as the rise divided by the run: \[ \text{Slope} = \frac{\text{Rise}}{\text{Run}} = \frac{4}{5} \]

So, the answers are:

  • Rise = 4
  • Run = 5
  • Slope = \(\frac{4}{5}\)
length(x) weight (y)
_______________
| 6 | 15 |
| 9 | 22.5 |
| 12. | 30 |
|. 15 | 37.5 |
the weight of the pet ferret varies directly with its length
the constant of proportionality is __
the following equation may be used to determine the weight of ferret given its length: y=__x

To determine the constant of proportionality and the equation relating the weight \( y \) to the length \( x \) for the pet ferret, we can analyze the provided data.

From the given length and weight values:

  • Length \( (x) \) and Weight \( (y) \):
    • \( x = 6, , y = 15 \)
    • \( x = 9, , y = 22.5 \)
    • \( x = 12, , y = 30 \)
    • \( x = 15, , y = 37.5 \)

To find the constant of proportionality, we can use any pair of \( (x, y) \) values. The constant of proportionality \( k \) can be found with the formula: \[ k = \frac{y}{x} \]

Using the first pair \( (6, 15) \): \[ k = \frac{15}{6} = 2.5 \]

We can verify this with the other pairs:

  • For \( x = 9, , y = 22.5 \): \[ k = \frac{22.5}{9} = 2.5 \]

  • For \( x = 12, , y = 30 \): \[ k = \frac{30}{12} = 2.5 \]

  • For \( x = 15, , y = 37.5 \): \[ k = \frac{37.5}{15} = 2.5 \]

Since \( k \) is consistently \( 2.5 \), the constant of proportionality is 2.5.

Next, we can write the equation relating weight \( y \) to length \( x \): \[ y = kx \] Substituting the value of \( k \): \[ y = 2.5x \]

So, the answers are:

  • The constant of proportionality is 2.5.
  • The equation may be used to determine the weight of the ferret given its length: \( y = 2.5x \).
select all of the expressions that are equivalent to this algebraic expression
(3x-8) + (2x+5) - (4x-8)
a. 3x-8+2x+5-4x-8
b. 3x-8+2x+5-4x+8
c. 3x+2x-4x-8+5+8
d. 5x-4x-8+5+8
e.x-8+5+8
f.x-13+8
g.x-3+8
h.x-21
i.x+5

To determine which expressions are equivalent to the algebraic expression \((3x - 8) + (2x + 5) - (4x - 8)\), we will first simplify this expression step by step.

Step 1: Simplify the original expression

  1. Expand the original expression: \[ (3x - 8) + (2x + 5) - (4x - 8) \] This becomes: \[ 3x - 8 + 2x + 5 - 4x + 8. \]

Step 2: Combine like terms

  1. Combine the \(x\) terms: \[ 3x + 2x - 4x = (3 + 2 - 4)x = 1x = x. \]
  2. Combine the constant terms: \[ -8 + 5 + 8 = (-8 + 8) + 5 = 0 + 5 = 5. \]

Final simplified expression:

Combining both results, we have: \[ x + 5. \]

Step 3: Check each option

Now we will evaluate each of the provided options to see which one(s) simplify to \(x + 5\).

a. \(3x - 8 + 2x + 5 - 4x - 8\)
Simplifying this yields: \[ (3x + 2x - 4x) + (-8 - 8 + 5) = x + (-16 + 5) = x - 11 \quad \text{(not equivalent)} \]

b. \(3x - 8 + 2x + 5 - 4x + 8\)
Simplifying this yields: \[ (3x + 2x - 4x) + (-8 + 8 + 5) = x + 5 \quad \text{(equivalent)} \]

c. \(3x + 2x - 4x - 8 + 5 + 8\)
Simplifying this yields: \[ (3x + 2x - 4x) + (-8 + 5 + 8) = x + 5 \quad \text{(equivalent)} \]

d. \(5x - 4x - 8 + 5 + 8\)
Simplifying this yields: \[ (5x - 4x) + (-8 + 5 + 8) = x + 5 \quad \text{(equivalent)} \]

e. \(x - 8 + 5 + 8\)
Simplifying this yields: \[ x + (-8 + 5 + 8) = x + 5 \quad \text{(equivalent)} \]

f. \(x - 13 + 8\)
Simplifying this yields: \[ x + (-13 + 8) = x - 5 \quad \text{(not equivalent)} \]

g. \(x - 3 + 8\)
Simplifying this yields: \[ x + (-3 + 8) = x + 5 \quad \text{(equivalent)} \]

h. \(x - 21\)
This is not equivalent to \(x + 5\) \quad \text{(not equivalent)}

i. \(x + 5\)
This is equivalent \quad \text{(equivalent)}

Conclusion:

The equivalent expressions to \((3x - 8) + (2x + 5) - (4x - 8)\) are:

  • b. \(3x - 8 + 2x + 5 - 4x + 8\)
  • c. \(3x + 2x - 4x - 8 + 5 + 8\)
  • d. \(5x - 4x - 8 + 5 + 8\)
  • e. \(x - 8 + 5 + 8\)
  • g. \(x - 3 + 8\)
  • i. \(x + 5\)

Thus, the selected options are: b, c, d, e, g, i.

correctly select the order of the steps to solve this equation
1/3(6x-9)=-33
step 1:
step 2:
step 3:
a. divide by 1/3 on both sides of the equation
b. add 1/3 to both sides of the equation
c. subtract 1/3 from both sides of the equation
d. multiply by 1/3 on both sides of the equation
e. subtract 6 from both sides of the equation
f. add 9 to both sides of the equation
g. subtract 9 from both sides of the equation
h. divide by 6 on both sides of the equation
i. multiply by 6 on both sides of the equation

To solve the equation \( \frac{1}{3}(6x - 9) = -33 \), we need to follow the proper order of operations. Here is the correct step-by-step method:

Step 1: Eliminate the fraction

To eliminate the fraction, we will multiply both sides of the equation by \( 3 \) (the reciprocal of \( \frac{1}{3} \)). This is equivalent to saying we multiply both sides by \( 3 \).

  • The relevant choice here is not directly listed but implies a multiplication by \( 3 \) which would correspond to d if it were phrased better.

Step 2: Simplify the equation

After multiplying by \( 3 \), the equation becomes: \[ 6x - 9 = -99 \]

Step 3: Isolate the variable

Next, we want to isolate the term with \( x \). First, we add \( 9 \) to both sides of the equation to remove \( -9 \): \[ 6x = -99 + 9 \]

Step 4: Solve for \( x \)

Now we simplify to: \[ 6x = -90 \] Finally, we divide by \( 6 \) to solve for \( x \): \[ x = -15 \]

Summary of Correct Steps

Based on the process described, the correct selection of steps is:

  1. Step 1: \( \text{a. multiply by 3 on both sides of the equation} \)
  2. Step 2: \( \text{f. add 9 to both sides of the equation} \)
  3. Step 3: \( \text{h. divide by 6 on both sides of the equation} \)

Correct Order of Steps

Since multiplying by \( \frac{1}{3} \) (from the original term) would create confusion, it might be interpreted differently; therefore, the best representation for direct operation based on choices would be to:

  • Start with multiplying by 3 (eliminating the \( \frac{1}{3} \)): (not directly listed)
  • Then add \( 9 \) (to correct \( -9 \))
  • Finally divide by \( 6 \)

So the precise order according to provided choices based on those corrections interpreting the "multiply by \( 3 \) " implies the need of the other options.

In terms of how to identify them, as choosing \( \text{d} \) would hint not directly, the options might best be given something simply clarified away from it.

However, from the list, the steps must be adapted on clarity for other resolving as:

  • \( d \) multiplying would rectify solving to \( g \).

So final resulting:

  1. d
  2. f
  3. h

Feel free to adjust any misunderstanding from how these imply logical application steps if the representation differs!