40. $x - \frac{1}{5} = -\frac{4}{5}$
41. $\frac{n}{7} = 4.5$
42. $7t - 20 = -50 - 8t$
43. $\frac{1}{2}(6c + 2) = -3(c - 1)$
44. find the area of the trapezoid.
(image of trapezoid with top base 9 cm, height 10 cm, bottom base 12 cm)
45. you type 168 words in $3\frac{1}{2}$ minutes. how many words do you type per minute?
46. order the values from least to greatest.
$|-32|, 22, -16, -|21|, |-10|$
47. you want to find the height of a drop tower at an amusement park. you take the measurements shown in the diagram. the right triangles created by each object and its shadow are similar. how tall is the drop tower?
(image of drop tower and a person with shadow: person is 5 ft tall, shadow 3 ft; drop tower shadow 84 ft)

Question

Accepted Answer

### Problem 40:
# Explanation:
## Step1: Add $\frac{1}{5}$ to both sides
To solve for $ x $ in the equation $ x - \frac{1}{5} = -\frac{4}{5} $, we add $\frac{1}{5}$ to both sides of the equation. This is based on the addition property of equality, which states that if we add the same number to both sides of an equation, the equality still holds.

$$
x - \frac{1}{5} + \frac{1}{5} = -\frac{4}{5} + \frac{1}{5}
$$

## Step2: Simplify both sides
Simplifying the left side, $ x - \frac{1}{5} + \frac{1}{5} $ simplifies to $ x $ because the $-\frac{1}{5}$ and $+\frac{1}{5}$ cancel each other out.

Simplifying the right side, we have $-\frac{4}{5} + \frac{1}{5}$. When adding fractions with the same denominator, we add the numerators and keep the denominator the same. So, $-4 + 1 = -3$, and the denominator is 5. Thus, $-\frac{4}{5} + \frac{1}{5} = -\frac{3}{5}$.

# Answer:
$ x = -\frac{3}{5} $

### Problem 41:
# Explanation:
## Step1: Multiply both sides by 7
To solve for $ n $ in the equation $ \frac{n}{7} = 4.5 $, we multiply both sides of the equation by 7. This is based on the multiplication property of equality, which states that if we multiply both sides of an equation by the same non - zero number, the equality still holds.

$$
7\times\frac{n}{7}=4.5\times7
$$

## Step2: Simplify both sides
Simplifying the left side, $ 7\times\frac{n}{7} $ simplifies to $ n $ because 7 and the denominator 7 cancel each other out.

Simplifying the right side, we calculate $ 4.5\times7 $. $ 4.5\times7 = 31.5 $.

# Answer:
$ n = 31.5 $

### Problem 42:
# Explanation:
## Step1: Add $ 8t $ to both sides
To solve for $ t $ in the equation $ 7t - 20=-50 - 8t $, we first add $ 8t $ to both sides of the equation. This is to get all the terms with $ t $ on one side.

$$
7t - 20+8t=-50 - 8t + 8t
$$

Simplifying both sides: the left side becomes $ 15t-20 $ (since $ 7t + 8t=15t $) and the right side becomes $-50$ (since $-8t + 8t = 0$). So we have $ 15t-20=-50 $.

## Step2: Add 20 to both sides
Next, we add 20 to both sides of the equation $ 15t-20=-50 $ to isolate the term with $ t $.

$$
15t-20 + 20=-50+20
$$

Simplifying both sides: the left side becomes $ 15t $ (since $-20 + 20 = 0$) and the right side becomes $-30$ (since $-50+20=-30$). So we have $ 15t=-30 $.

## Step3: Divide both sides by 15
Now, we divide both sides of the equation $ 15t=-30 $ by 15 to solve for $ t $.

$$
\frac{15t}{15}=\frac{-30}{15}
$$

Simplifying both sides: the left side becomes $ t $ and the right side becomes $-2$ (since $\frac{-30}{15}=-2$).

# Answer:
$ t=-2 $

### Problem 43:
# Explanation:
## Step1: Distribute on both sides
First, we distribute the $\frac{1}{2}$ on the left side of the equation $\frac{1}{2}(6c + 2)=-3(c - 1)$ and the $-3$ on the right side.

For the left side: $\frac{1}{2}(6c+2)=\frac{1}{2}\times6c+\frac{1}{2}\times2 = 3c + 1$

For the right side: $-3(c - 1)=-3\times c-3\times(-1)=-3c + 3$

So the equation becomes $ 3c + 1=-3c + 3 $

## Step2: Add $ 3c $ to both sides
To get all the $ c $ terms on one side, we add $ 3c $ to both sides of the equation $ 3c + 1=-3c + 3 $

$$
3c + 1+3c=-3c + 3+3c
$$

Simplifying both sides: the left side becomes $ 6c + 1 $ (since $ 3c+3c = 6c $) and the right side becomes $ 3 $ (since $-3c + 3c = 0$). So we have $ 6c+1 = 3 $

## Step3: Subtract 1 from both sides
Next, we subtract 1 from both sides of the equation $ 6c+1 = 3 $ to isolate the term with $ c $.

$$
6c+1 - 1=3 - 1
$$

Simplifying both sides: the left side becomes $ 6c $ (since $ 1-1 = 0$) and the right side becomes $ 2 $ (since $ 3 - 1=2$). So we have $ 6c=2 $

## Step4: Divide both sides by 6
Now, we divide both sides of the equation $ 6c=2 $ by 6 to solve for $ c $.

$$
c=\frac{2}{6}=\frac{1}{3}
$$

# Answer:
$ c=\frac{1}{3} $

### Problem 44:
# Explanation:
## Step1: Recall the formula for the area of a trapezoid
The formula for the area $ A $ of a trapezoid is $ A=\frac{(a + b)h}{2} $, where $ a $ and $ b $ are the lengths of the two parallel sides (the bases) and $ h $ is the height (the perpendicular distance between the two bases).

In the given trapezoid, $ a = 9\space cm $, $ b = 12\space cm $, and $ h = 10\space cm $.

## Step2: Substitute the values into the formula
Substitute $ a = 9 $, $ b = 12 $, and $ h = 10 $ into the formula $ A=\frac{(a + b)h}{2} $

First, calculate $ a + b $: $ 9+12 = 21 $

Then, multiply by $ h $: $ 21\times10=210 $

Then, divide by 2: $ A=\frac{210}{2}=105 $

# Answer:
The area of the trapezoid is $ 105\space cm^{2} $

### Problem 45:
# Explanation:
## Step1: Recall the formula for rate
The rate of typing (words per minute) is given by the formula $ \text{Rate}=\frac{\text{Number of words}}{\text{Time in minutes}} $

We know that the number of words is 168 and the time is $ 3\frac{1}{2}$ minutes. We need to convert $ 3\frac{1}{2}$ to an improper fraction. $ 3\frac{1}{2}=\frac{3\times2 + 1}{2}=\frac{7}{2} $

## Step2: Calculate the rate
Substitute the number of words (168) and time ($\frac{7}{2}$ minutes) into the rate formula.

$ \text{Rate}=168\div\frac{7}{2} $

When dividing by a fraction, we multiply by its reciprocal. So $ 168\div\frac{7}{2}=168\times\frac{2}{7} $

First, divide 168 by 7: $ 168\div7 = 24 $

Then, multiply by 2: $ 24\times2=48 $

# Answer:
You type 48 words per minute.

### Problem 46:
# Explanation:
## Step1: Calculate the absolute values
First, we calculate the absolute values of the given numbers:

- For $|-32|$, the absolute value of a number is its distance from 0 on the number line. So $|-32| = 32$
- For $-|21|$, first calculate $|21|=21$, then apply the negative sign, so $-|21|=-21$
- For $|-10|$, $|-10| = 10$

So the numbers we have to order are: 32, 22, - 16, - 21, 10

## Step2: Order the numbers from least to greatest
To order these numbers, we know that negative numbers are less than positive numbers. Among the negative numbers, the larger the absolute value, the smaller the number.

So, $-21$ (which is $-|21|$) is less than $-16$ because $|-21| = 21$ and $|-16| = 16$ and $21>16$.

Then, among the positive numbers, $10$ (which is $|-10|$) is less than $22$ which is less than $32$ (which is $|-32|$)

So the order from least to greatest is: $-|21|,-16,|-10|,22,|-32|$

# Answer:
$-|21|,-16,|-10|,22,|-32|$

### Problem 47:
# Explanation:
## Step1: Set up the proportion
Since the two right triangles (one formed by the drop tower and its shadow, and the other formed by the 5 - ft object and its 3 - ft shadow) are similar, the ratios of their corresponding sides are equal.

Let $ h $ be the height of the drop tower. The ratio of the height of the drop tower to the length of its shadow is $ \frac{h}{84} $, and the ratio of the height of the 5 - ft object to the length of its shadow is $ \frac{5}{3} $

Since the triangles are similar, we can set up the proportion:

$$
\frac{h}{84}=\frac{5}{3}
$$

## Step2: Solve for $ h $
To solve for $ h $, we cross - multiply.

$ 3h=84\times5 $

First, calculate $ 84\times5 = 420 $

Then, divide both sides by 3: $ h=\frac{420}{3}=140 $

# Answer:
The height of the drop tower is 140 feet.

Sovi.AI - AI Math Tutor

QUESTION IMAGE

Question

Explanation:

Problem 40:

Step1: Add $\frac{1}{5}$ to both sides

Step2: Simplify both sides

Step1: Multiply both sides by 7

Step2: Simplify both sides

Step1: Add \( 8t \) to both sides

Step2: Add 20 to both sides

Step3: Divide both sides by 15

Answer:

Problem 41: