* 16. connect there are 100° on the celsius scale from the freezing temperature to the boiling temperature of water. there are 180° on the fahrenheit scale between these temperatures. so a change in temperature of 10° on the celsius scale is equivalent to a change of how many degrees on the fahrenheit scale?
solve:
17. $\frac{3}{2.5} = \frac{48}{c}$
18. $k - 0.75 = 0.75$
simplify:
19. $15^2 - 5^3 - \sqrt{100}$
20. $6 + 12 \div 3 \cdot 2 - 3 \cdot 4$
21. $\begin{array}{r} 5\text{ yd } 2\text{ ft } 3\text{ in.} \\ + 2\text{ yd } 2\text{ ft } 9\text{ in.} \\ \hline \end{array}$
* 22. $\begin{array}{r} 5\text{ yd } 2\text{ ft } 3\text{ in.} \\ - 2\text{ yd } 2\text{ ft } 9\text{ in.} \\ \hline \end{array}$
23. $\frac{18}{19} \cdot \frac{19}{18}$
24. $2\frac{3}{4} + \left(5\frac{1}{6} - 1\frac{1}{4}\
ight)$
25. $3\frac{3}{4} \cdot 2\frac{1}{2} \div 3\frac{1}{8}$
26. $3\frac{3}{4} \div 2\frac{1}{2} \cdot 3\frac{1}{8}$
27. describe how to find the 99th number in this sequence: $1, 4, 9, 16, 25, \dots$
28. use a protractor and a straightedge to draw a triangle that has a right angle and a 30° angle. then measure the shortest and longest sides of the triangle to the nearest millimeter. what is the relationship of the two measurements?
29. if the diameter of a wheel is 0.5 meter, then the radius of the wheel is how many centimeters?
* 30. represent the diameter of a circle is a function of the radius of the circle, expressed by the equation $d = 2r$.
a. make a function table for $r$ (input) and $d$ (output). record three pairs in the table using small whole numbers for the values for $r$.
b. on a coordinate plane, label the horizontal axis $r$ and the vertical axis $d$. then graph all the first quadrant pairs that satisfy the function.
c. discuss how do we indicate that our graph includes all of the points in the first quadrant that satisfy the function?

Question

* 16. connect there are 100° on the celsius scale from the freezing temperature to the boiling temperature of water. there are 180° on the fahrenheit scale between these temperatures. so a change in temperature of 10° on the celsius scale is equivalent to a change of how many degrees on the fahrenheit scale?
solve:
17. $\frac{3}{2.5} = \frac{48}{c}$
18. $k - 0.75 = 0.75$
simplify:
19. $15^2 - 5^3 - \sqrt{100}$
20. $6 + 12 \div 3 \cdot 2 - 3 \cdot 4$
21. $\begin{array}{r} 5\text{ yd } 2\text{ ft } 3\text{ in.} \\ + 2\text{ yd } 2\text{ ft } 9\text{ in.} \\ \hline \end{array}$
* 22. $\begin{array}{r} 5\text{ yd } 2\text{ ft } 3\text{ in.} \\ - 2\text{ yd } 2\text{ ft } 9\text{ in.} \\ \hline \end{array}$
23. $\frac{18}{19} \cdot \frac{19}{18}$
24. $2\frac{3}{4} + \left(5\frac{1}{6} - 1\frac{1}{4}\
ight)$
25. $3\frac{3}{4} \cdot 2\frac{1}{2} \div 3\frac{1}{8}$
26. $3\frac{3}{4} \div 2\frac{1}{2} \cdot 3\frac{1}{8}$
27. describe how to find the 99th number in this sequence: $1, 4, 9, 16, 25, \dots$
28. use a protractor and a straightedge to draw a triangle that has a right angle and a 30° angle. then measure the shortest and longest sides of the triangle to the nearest millimeter. what is the relationship of the two measurements?
29. if the diameter of a wheel is 0.5 meter, then the radius of the wheel is how many centimeters?
* 30. represent the diameter of a circle is a function of the radius of the circle, expressed by the equation $d = 2r$.
    a. make a function table for $r$ (input) and $d$ (output). record three pairs in the table using small whole numbers for the values for $r$.
    b. on a coordinate plane, label the horizontal axis $r$ and the vertical axis $d$. then graph all the first quadrant pairs that satisfy the function.
    c. discuss how do we indicate that our graph includes all of the points in the first quadrant that satisfy the function?

Accepted Answer

### Problem 16
# Explanation:
## Step1: Find the ratio of Fahrenheit to Celsius degrees change.
The change in Celsius is $100^\circ$ and in Fahrenheit is $180^\circ$, so the ratio is $\frac{180}{100}$.

## Step2: Calculate the equivalent Fahrenheit change for $10^\circ$ Celsius.
Multiply the Celsius change ($10^\circ$) by the ratio: $10\times\frac{180}{100}$.

## Step3: Simplify the expression.
$10\times\frac{180}{100} = \frac{1800}{100}=18$.

# Answer:
$18^\circ$ Fahrenheit

### Problem 17
# Explanation:
## Step1: Cross - multiply the proportion $\frac{3}{2.5}=\frac{48}{c}$.
Using the cross - multiplication property of proportions ($a/b = c/d$ implies $a\times d=b\times c$), we get $3\times c = 2.5\times48$.

## Step2: Calculate the right - hand side.
$2.5\times48 = 120$, so the equation becomes $3c = 120$.

## Step3: Solve for $c$.
Divide both sides of the equation by 3: $c=\frac{120}{3}=40$.

# Answer:
$c = 40$

### Problem 18
# Explanation:
## Step1: Isolate the variable $k$.
To solve for $k$ in the equation $k - 0.75=0.75$, we add $0.75$ to both sides of the equation.

## Step2: Calculate the value of $k$.
$k=0.75 + 0.75=1.5$.

# Answer:
$k = 1.5$

### Problem 19
# Explanation:
## Step1: Calculate each term.
- Calculate $15^{2}$: $15^{2}=15\times15 = 225$.
- Calculate $5^{3}$: $5^{3}=5\times5\times5 = 125$.
- Calculate $\sqrt{100}$: $\sqrt{100}=10$.

## Step2: Substitute the values into the expression.
The expression $15^{2}-5^{3}-\sqrt{100}$ becomes $225-125 - 10$.

## Step3: Perform the subtraction.
First, $225-125 = 100$, then $100 - 10=90$.

# Answer:
$90$

### Problem 20
# Explanation:
## Step1: Follow the order of operations (PEMDAS: Parentheses, Exponents, Multiplication and Division from left to right, Addition and Subtraction from left to right).
First, perform the division: $12\div3 = 4$.

## Step2: Perform the multiplication.
Then, $4\times2 = 8$ and $3\times4 = 12$.

## Step3: Perform the addition and subtraction.
The expression $6 + 12\div3\times2-3\times4$ becomes $6 + 8-12$. First, $6 + 8=14$, then $14-12 = 2$.

# Answer:
$2$

### Problem 21
# Explanation:
## Step1: Add the inches.
$3$ in $+9$ in $ = 12$ in. Since $12$ in $ = 1$ ft, we carry over $1$ ft and have $0$ in left.

## Step2: Add the feet.
$2$ ft $+2$ ft $+1$ ft (carry - over) $ = 5$ ft. Since $5$ ft $ = 1$ yd and $2$ ft left (because $5\div3 = 1$ yd with a remainder of $2$ ft), we carry over $1$ yd.

## Step3: Add the yards.
$5$ yd $+2$ yd $+1$ yd (carry - over) $ = 8$ yd.

## Step4: Combine the results.
So the sum is $8$ yd $2$ ft $0$ in.

# Answer:
$8$ yd $2$ ft $0$ in

### Problem 22
# Explanation:
## Step1: Subtract the inches.
We can't subtract $9$ in from $3$ in, so we borrow $1$ ft from the feet column. $1$ ft $ = 12$ in, so $3$ in $+12$ in $ = 15$ in. Then $15$ in $-9$ in $ = 6$ in.

## Step2: Subtract the feet.
After borrowing $1$ ft, the number of feet is $2 - 1=1$ ft. Now subtract the feet: $1$ ft $-2$ ft. We can't do this, so we borrow $1$ yd from the yards column. $1$ yd $ = 3$ ft, so $1$ ft $+3$ ft $ = 4$ ft. Then $4$ ft $-2$ ft $ = 2$ ft.

## Step3: Subtract the yards.
After borrowing $1$ yd, the number of yards is $5 - 1 = 4$ yd. Then $4$ yd $-2$ yd $ = 2$ yd.

## Step4: Combine the results.
So the difference is $2$ yd $2$ ft $6$ in.

# Answer:
$2$ yd $2$ ft $6$ in

### Problem 23
# Explanation:
## Step1: Cancel out the common factors.
Notice that $18$ in the numerator of the first fraction and $18$ in the denominator of the second fraction cancel out, and $19$ in the denominator of the first fraction and $19$ in the numerator of the second fraction cancel out.

## Step2: Calculate the product.
After canceling, we have $\frac{1}{1}\times\frac{1}{1}=1$.

# Answer:
$1$

### Problem 24
# Explanation:
## Step1: Subtract the mixed numbers inside the parentheses.
First, convert the mixed numbers to improper fractions: $5\frac{1}{6}=\frac{31}{6}$ and $1\frac{1}{4}=\frac{5}{4}$. Find a common denominator, which is $12$. $\frac{31}{6}=\frac{62}{12}$ and $\frac{5}{4}=\frac{15}{12}$. Then $\frac{62}{12}-\frac{15}{12}=\frac{47}{12}=3\frac{11}{12}$.

## Step2: Add the remaining mixed number.
$2\frac{3}{4}=\frac{11}{4}=\frac{33}{12}$. Now add $\frac{33}{12}+ \frac{47}{12}=\frac{80}{12}=\frac{20}{3}=6\frac{2}{3}$.

# Answer:
$6\frac{2}{3}$

### Problem 25
# Explanation:
## Step1: Convert mixed numbers to improper fractions.
$3\frac{3}{4}=\frac{15}{4}$, $2\frac{1}{2}=\frac{5}{2}$, $3\frac{1}{8}=\frac{25}{8}$.

## Step2: Perform the multiplication and division from left to right.
First, multiply $\frac{15}{4}\times\frac{5}{2}=\frac{75}{8}$. Then divide by $\frac{25}{8}$: $\frac{75}{8}\div\frac{25}{8}=\frac{75}{8}\times\frac{8}{25}=3$.

# Answer:
$3$

### Problem 26
# Explanation:
## Step1: Convert mixed numbers to improper fractions.
$3\frac{3}{4}=\frac{15}{4}$, $2\frac{1}{2}=\frac{5}{2}$, $3\frac{1}{8}=\frac{25}{8}$.

## Step2: Perform the division and multiplication from left to right.
First, divide $\frac{15}{4}\div\frac{5}{2}=\frac{15}{4}\times\frac{2}{5}=\frac{3}{2}$. Then multiply by $\frac{25}{8}$: $\frac{3}{2}\times\frac{25}{8}=\frac{75}{16}=4\frac{11}{16}$.

# Answer:
$4\frac{11}{16}$

### Problem 27
# Explanation:
## Step1: Identify the pattern of the sequence.
The sequence $1,4,9,16,25,\cdots$ is the sequence of perfect squares. The $n$th term of the sequence is given by $a_{n}=n^{2}$, where $n = 1,2,3,\cdots$.

## Step2: Find the 99th term.
To find the 99th term, we substitute $n = 99$ into the formula $a_{n}=n^{2}$. So $a_{99}=99^{2}$.

## Step3: Calculate $99^{2}$.
$99^{2}=(100 - 1)^{2}=100^{2}-2\times100\times1 + 1^{2}=10000-200 + 1 = 9801$.

# Answer:
The 99th term is $9801$ (found by squaring 99, since the $n$th term of the sequence $1,4,9,16,25,\cdots$ is $n^{2}$).

### Problem 28
# Explanation:
## Step1: Draw the right - angled triangle with a $30^{\circ}$ angle.
- Use the straightedge to draw a horizontal line segment.
- Place the protractor on one end of the horizontal line segment, mark a $90^{\circ}$ angle (for the right angle) and a $30^{\circ}$ angle.
- Draw the other two sides of the triangle using the straightedge.

## Step2: Measure the sides.
- Let the length of the side opposite the $30^{\circ}$ angle be $a$, the side opposite the $60^{\circ}$ angle (since the angles in a triangle sum to $180^{\circ}$, the third angle is $180 - 90 - 30=60^{\circ}$) be $b$, and the hypotenuse (opposite the right angle) be $c$.
- In a $30 - 60 - 90$ triangle, the ratio of the sides is $a:b:c = 1:\sqrt{3}:2$. So the length of the hypotenuse $c$ is twice the length of the side opposite the $30^{\circ}$ angle ($a$), i.e., $c = 2a$.

# Answer:
The hypotenuse (longest side) is twice the length of the side opposite the $30^{\circ}$ angle (shortest non - hypotenuse side).

### Problem 29
# Explanation:
## Step1: Recall the relationship between diameter and radius.
The radius $r$ of a circle is half of the diameter $d$, so $r=\frac{d}{2}$.

## Step2: Calculate the radius in meters.
Given $d = 0.5$ m, then $r=\frac{0.5}{2}=0.25$ m.

## Step3: Convert meters to centimeters.
Since $1$ m $ = 100$ cm, then $0.25$ m $=0.25\times100 = 25$ cm.

# Answer:
$25$ centimeters

### Problem 30
#### Part (a)
# Explanation:
## Step1: Choose values for $r$.
Let's choose $r = 1$, $r = 2$, and $r = 3$ (small whole numbers).

## Step2: Calculate $d$ for each $r$ using $d = 2r$.
- When $r = 1$, $d=2\times1 = 2$.
- When $r = 2$, $d=2\times2 = 4$.
- When $r = 3$, $d=2\times3 = 6$.

## Step3: Create the function table.
| $r$ (input) | $d$ (output) |
|---|---|
| $1$ | $2$ |
| $2$ | $4$ |
| $3$ | $6$ |

# Answer:
The function table with three pairs $(1,2)$, $(2,4)$, $(3,6)$ (as shown above).

#### Part (b)
# Explanation:
## Step1: Set up the coordinate plane.
Label the horizontal axis as $r$ and the vertical axis as $d$.

## Step2: Plot the points.
- For $r = 1$, $d = 2$, plot the point $(1,2)$.
- For $r = 2$, $d = 4$, plot the point $(2,4)$.
- For $r = 3$, $d = 6$, plot the point $(3,6)$.

## Step3: Draw the line.
Since the function $d = 2r$ is a linear function, draw a straight line passing through these points and extending in the first quadrant (since $r$ and $d$ represent lengths and are non - negative).

# Answer:
A straight line passing through the points $(1,2)$, $(2,4)$, $(3,6)$ (and other points satisfying $d = 2r$) in the first quadrant of the coordinate plane with $r$ on the x - axis and $d$ on the y - axis.

#### Part (c)
# Explanation:
## Step1: Understand the nature of the function.
The function $d = 2r$ is a linear function where $r$ can take any non - negative real number (since $r$ represents the radius of a circle, it must be non - negative).

## Step2: Indicate all points.
To indicate that the graph includes all of the points in the first quadrant that satisfy the function, we draw an arrow on the end of the line we drew in part (b) (extending the line infinitely in the direction of increasing $r$ and $d$ in the first quadrant) and also, since the relationship is linear and holds for all non - negative real numbers $r$, the line represents all possible pairs $(r,d)$ where $d = 2r$ and $r\geq0$, $d\geq0$.

# Answer:
We draw an arrow on the line (extending it infinitely in the first quadrant) to show that it includes all points $(r,d)$ with $r\geq0$ and $d = 2r$ (all non - negative real - valued input - output pairs that satisfy $d = 2r$).

Sovi.AI - AI Math Tutor

QUESTION IMAGE

Question

Explanation:

Problem 16

Step1: Find the ratio of Fahrenheit to Celsius degrees change.

Step2: Calculate the equivalent Fahrenheit change for \(10^\circ\) Celsius.

Step3: Simplify the expression.

Step1: Cross - multiply the proportion \(\frac{3}{2.5}=\frac{48}{c}\).

Step2: Calculate the right - hand side.

Step3: Solve for \(c\).

Step1: Isolate the variable \(k\).

Step2: Calculate the value of \(k\).

Answer:

Problem 17