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Using the binomial expansion theorem, the values of integers p and q are -8 and 24, respectively

From the question, we are to use the binomial expansion to expand the given expression and determine the values of p and q.

We can expand (1 - 2x)^4 using the binomial theorem as follows:

(1 - 2x)^4 = 1^4 - 4(1^3)(2x) + 6(1^2)(2x)^2 - 4(1)(2x)^3 + (2x)^4

= 1 - 8x + 24x^2 - 32x^3 + 16x^4

Now, we will compare this expression to the given expression

Comparing the expression to the given expression, 1 + px + qx^2 - 32x^3 + 16x^4

We see that:

p = -8

q = 24

Hence, the values p and q are -8 and 24, respectively.

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There are 6 + 9 + 9 + 12 + 9 + 9 + 6 = 60 outcomes where the sum of the three dice is greater than 5.

So, the probability of rolling a sum greater than 5 is given by:

Probability = Number of favorable outcomes / Total number of outcomes
Probability = 60 / 216
Probability = 5 / 18

We know that when a dice is rolled, the numbers that come up on the dice are 1, 2, 3, 4, 5 and 6. Since there are three dice, the total number of possible outcomes when they are tossed is given by 6 * 6 * 6 = 216.

Now we have to find the probability of rolling a sum greater than 5. To find this probability, we need to consider all the cases where the sum of the three dice is greater than 5.

The possible outcomes where the sum of the three dice is greater than 5 are:

Sum of 6: (1, 2, 3), (1, 3, 2), (2, 1, 3), (2, 3, 1), (3, 1, 2), (3, 2, 1) (Total 6)
Sum of 7: (1, 2, 4), (1, 4, 2), (2, 1, 4), (2, 4, 1), (4, 1, 2), (4, 2, 1), (1, 3, 3), (3, 1, 3), (3, 3, 1) (Total 9)
Sum of 8: (1, 2, 5), (1, 5, 2), (2, 1, 5), (2, 5, 1), (5, 1, 2), (5, 2, 1), (3, 3, 2), (3, 2, 3), (2, 3, 3) (Total 9)
Sum of 9: (1, 3, 5), (1, 5, 3), (3, 1, 5), (3, 5, 1), (5, 1, 3), (5, 3, 1), (4, 2, 3), (4, 3, 2), (2, 4, 3), (3, 4, 2), (2, 3, 4), (3, 2, 4) (Total 12)
Sum of 10: (1, 4, 5), (1, 5, 4), (4, 1, 5), (4, 5, 1), (5, 1, 4), (5, 4, 1), (2, 4, 4), (4, 2, 4), (4, 4, 2) (Total 9)
Sum of 11: (1, 5, 5), (5, 1, 5), (5, 5, 1), (2, 5, 4), (2, 4, 5), (4, 5, 2), (4, 2, 5), (5, 4, 2), (5, 2, 4) (Total 9)
Sum of 12: (3, 4, 5), (3, 5, 4), (4, 3, 5), (4, 5, 3), (5, 3, 4), (5, 4, 3) (Total 6)

Therefore, there are 6 + 9 + 9 + 12 + 9 + 9 + 6 = 60 outcomes where the sum of the three dice is greater than 5.

So, the probability of rolling a sum greater than 5 is given by:

Probability = Number of favorable outcomes / Total number of outcomes
Probability = 60 / 216
Probability = 5 / 18

Hence, the probability of rolling a sum greater than 5 is 5 / 18.

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The area of the triangle is 715.7 square units, rounded off to one decimal place.

The given triangle's side lengths and the angles are a = 52, B = 115°, and C = 29°. The area of the triangle can be determined by applying the formula:A = (1/2) a² sin B sin C, where a is the length of the side opposite to angle A.The area of the triangle is  (rounding off to one decimal place)Therefore, the area of the triangle is 715.7 square units, rounded off to one decimal place.

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Answer:

14, 7, 0, -7.

Step-by-step explanation:

To find the first four terms of the sequence, we can substitute different values of n into the given expression and simplify.

The expression for the nth term of the sequence is 7(3 - n).

Let's find the value of the first term (n = 1):

T₁ = 7(3 - 1) = 7(2) = 14

The first term of the sequence is 14.

Now, let's find the value of the second term (n = 2):

T₂ = 7(3 - 2) = 7(1) = 7

The second term of the sequence is 7.

Next, let's find the value of the third term (n = 3):

T₃ = 7(3 - 3) = 7(0) = 0

The third term of the sequence is 0.

Finally, let's find the value of the fourth term (n = 4):

T₄ = 7(3 - 4) = 7(-1) = -7

The fourth term of the sequence is -7.

Therefore, the first four terms of the sequence are:

To find the first four terms of the sequence, we can substitute different values of n into the given expression and simplify.

The expression for the nth term of the sequence is 7(3 - n).

Let's find the value of the first term (n = 1):

T₁ = 7(3 - 1) = 7(2) = 14

The first term of the sequence is 14.

Now, let's find the value of the second term (n = 2):

T₂ = 7(3 - 2) = 7(1) = 7

The second term of the sequence is 7.

Next, let's find the value of the third term (n = 3):

T₃ = 7(3 - 3) = 7(0) = 0

The third term of the sequence is 0.

Finally, let's find the value of the fourth term (n = 4):

T₄ = 7(3 - 4) = 7(-1) = -7

The fourth term of the sequence is -7.

Therefore, the first four terms of the sequence are:

14, 7, 0, -7.

Answer:

yes you are right

Step-by-step explanation:

move 4 to other side so its -3x=12

divide 12 by -3

x=-4

Step-by-step explanation:

use the function sohcahtoa to solve as all these triangles are right angled

Based on the coordinate plane, the ordered pair for the post office is (9, 9).

In Mathematics, an ordered pair is sometimes referred to as a coordinate and it can be defined as a pair of two (2) elements or data points that are commonly written in a fixed order within parentheses as (x, y), which represents the x-coordinate (abscissa) and the y-coordinate (ordinate) on the coordinate plane of any graph.

By critically observing the coordinate plane shown in the image attached below, we can logically deduce that the coordinates of the post office would be located in quadrant I at point (9, 9).

Similarly, the the coordinates of the library is also located in quadrant I and it is represented by the ordered pair (6, 2).

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For instance, f(x) is equal to 0.5 when x = 1 and equal to 1 when x = 2, function indicating that the amount of water collected rises by 0.5 for each unit increase in time.

Mathematicians research numbers, their variants, equations, forms, and related structures, as well as possible locations for these things. The relationship between a group of inputs, each of which has a corresponding output, is referred to as a function. Every input contributes to a single, distinct output in a connection between inputs and outputs known as a function. A domain, codomain, or scope is assigned to each function. Often, functions are denoted with the letter f. (x). The key is an x. There are four main categories of accessible functions: on functions, one-to-one capabilities, so many capabilities, in capabilities, and on functions.

A. It appears that the function f(x) = 2x is exponential.

b. The function may be expressed as f(x) = a * bx, where a denotes the starting value, b the growth factor, and x the input value. As can be seen from the table, the values of f(x) = 2x are exponentially growing by a factor of 2 for each input.

d. A linear function may be used to simulate a tub that collects water from a leaking faucet since the amount of water collected grows steadily over time.

For instance, f(x) is equal to 0.5 when x = 1 and equal to 1 when x = 2, indicating that the amount of water collected rises by 0.5 for each unit increase in time.

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The areas of each part of the composite figure are;

Triangle A = 20 Square units

Triangle B = 2 Square units

Rectangle C = 4 Square units

Triangle D = 6 Square units

Area is a measurement of the two-dimensional surface of a shape or object. Area is often used when measuring the size of a plot of land or other physical space, such as a room or an outdoor area.

The area of each part of the figure can be found by adding the areas of the individual shapes that make up the figure. The area of a triangle can be found by using the formula A = 1/2bh, where b is the base and h is the height of the triangle. For a rectangle, the area is equal to the length multiplied by the width.

The composite figure's component parts' respective areas are;

20 Square Units = Triangle A

Triangle B = 2 units of the square

Square units = 4 for the rectangle C.

Triangle D = 6 units of the square

The formula for a triangle's area is straightforward;

A = 0.5 × base × height

Triangle A's area is;

Triangle A: (6 + 2 + 2) × 4 × 1/2

= ¹/₂ × 10 × 4

equals 20 square units

Triangle B's perimeter is;

Triangle A equals 1/2 × 2 × 2

equals 2 square units

Length × Width = Area of Rectangle C

= 2 × 2

equals 4 square units

Triangle D's area is;

Triangle D is equal to.5 × 6.

equals 6 square units

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Complete question -

The simplified expression is 8x² + 24x.

The distributive property of multiplication over addition is a fundamental property of arithmetic that relates multiplication and addition. It states that when you multiply a number by the sum of two or more numbers, you can first distribute the multiplication over each addend and then perform the addition.

In other words, if a, b, and c are any numbers, then:

a x (b + c) = (a x b) + (a x c)

To simplify the expression 4x° (2x + 6)°, we can use the distributive property of multiplication over addition.

4x° (2x + 6)°

= 4x° * 2x° + 4x° * 6° (using distributive property)

= 8x² + 24x (simplifying by multiplying)

Therefore, the simplified expression is 8x² + 24x.

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hypotenuse^2 = 16^2 + 25^2

hypotenuse^2 = 256 + 625

hypotenuse^2 = 881

hypotenuse = sqrt(881)

hypotenuse ≈ 29.67 inches

Now that we know the length of the hypotenuse, we can use the 16-inch side and the hypotenuse as the base and height of the triangle, respectively. Plugging these values into the formula, we get:

Area = (16 x 29.67) / 2

Area ≈ 237.36 square inches

Therefore, the area of the triangle is approximately 237.36 square inches.

Answer:

  m∠2 = 35°

Step-by-step explanation:

You want to know the measure of angle 2 when angle 1 is 35° and both angles 1 and 2 intercept arc PQ.

Inscribed angles 1 and 2 both intercept the same arc: PQ. The problem statement tells you that such angles are equal.

  ∠2 = ∠1 = 35°

The measure of ∠2 is 35°.

__

Additional comment

This is a vocabulary and reading comprehension test.

In order to understand the comment and the question, you need to know the meaning of "inscribed angle", "intercept [an] arc", "equal" (as applied to angles). You also need to know the meaning of the notation m∠1, the measure of angle 1.

You pass the test when you understand the question is telling you that angles 1 and 2 are both 35°.

Z = X/Y is normally distributed because  the ratio of two independent normal variables is itself normally distributed and the same has been proved below:

To prove this, we can use the Central Limit Theorem. This theorem states that if X and Y are independently and identically distributed random variables, then the ratio of the two, Z = X/Y, will be normally distributed regardless of the distribution of X and Y. This is due to the fact that the ratio of two independent normal variables is itself normally distributed.

For example, let X and Y be two independent normal variables. Then their ratio Z = X/Y will follow a normal distribution. This means that the probability density function (pdf) of Z is given by:

f_Z(z) = \frac{1}{\sqrt{2\pi\sigma^2}}e^{-\frac{z^2}{2\sigma^2}}

where \sigma^2 = \frac{\sigma_x^2}{\sigma_y^2} is the variance of Z.

Therefore, we can conclude that Z = X/Y is normally distributed when X and Y are independent normal distributions.

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Based οn this pattern, we can determine the vοlume οf the next figure in the pattern by using a height οf 4 units: 1 x 1 x 4 = 4 cubic units.

A cube is three-dimensiοnal sοlid οbject that has six square faces οf equal size. It is special type οf rectangular prism in which all six faces are squares οf equal size. A cube has twelve edges οf equal length and eight vertices where three edges meet.

Let's analyze each figure separately and find the vοlume using the fοrmula fοr the vοlume οf a rectangular prism: length x width x height.

First figure: The length, width, and height are all 1 unit, sο the vοlume is 1 x 1 x 1 = 1 cubic unit.

Secοnd figure: The length and width are still 1 unit, but the height is nοw 2 units. Sο, the vοlume is 1 x 1 x 2 = 2 cubic units.

Third figure: The length and width are still 1 unit, but the height is nοw 3 units. Sο, the vοlume is 1 x 1 x 3 = 3 cubic units.

Frοm the analysis, we can see that the pattern is that the length and width οf each figure remain cοnstant at 1 unit, while the height increases by 1 unit fοr each successive figure.

Based οn this pattern, we can determine the vοlume οf the next figure in the pattern by using a height οf 4 units: 1 x 1 x 4 = 4 cubic units.

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The ratio x:y in its simplest form is 7:4.

The ratio x:y can be found by dividing x by y. In this case, we can start with the equation y = 4x/7 and solve for x in terms of y.

Multiplying both sides by 7, we get:

7y = 4x

Dividing both sides by 4, we get:

x = 7y/4

Now we can substitute this expression for x into the ratio x:y:

x:y = (7y/4):y

Simplifying by canceling out the y, we get:

x:y = 7/4

So the ratio x:y in its simplest form is 7:4.

This means that for every 7 units of x, there are 4 units of y. The ratio cannot be simplified further because 7 and 4 do not have any common factors other than 1.

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The toe of which the triangle is , is called an isosceles triangle and a right angled triangle.

A triangle is a closed, 2-dimensional shape with 3 sides, 3 angles, and 3 vertices. There are different types of triangle , some of them are ;

Scalene triangle, isosceles triangle , equilateral triangle e.tc.

To know the type of triangle it is, we need to find the length of each sides.

let A = (2,1)

B = (4,7)

C = ( 6,3)

AB = √ (4-2)²+ (7-1)²

AB = √ 2²+ 6²

AB = √2+36

AB = √40

= 2√10

BC = √ (6-4)²+( 3-7)²

BC = √ 2²+4²

BC = √4+16

BC = √20

= 2√5

AC = √ (6-2)²+(3-1)²

AC = √4²+2²

AC =√ 16+2

AC = √20

= 2√5

therefore since AB² = BC² + AC ² ,the triangle is a right angled triangle

And also since two sides of the triangle are equal it is an isosceles triangle.

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Answer:

To determine the best value between chocolate bar A and B, we need to calculate the cost per gram of each chocolate bar.

For chocolate bar A, the cost per gram is:

$1.00 ÷ 80 grams = $0.0125 per gram

For chocolate bar B, the cost per gram is:

$1.20 ÷ 85 grams = $0.0141 per gram

Therefore, chocolate bar A is the better value as it costs less per gram compared to chocolate bar B. While chocolate bar B may weigh slightly more, its higher cost per gram means that you are paying more for each gram of chocolate compared to chocolate bar A.

1. The upper bound for the perimeter of the regular hexagon is 21.9 cm.

2. All figures on the calculator display for the calculation of Kelly's average speed is: 99.5 / 10.51 = 9.46717412

3. the lower bound for the area of the rectangle is 2.55 × 10⁵ mm²

4. Upper bound for area = 937.6525 cm²

1. The upper bound for the perimeter of the regular hexagon can be calculated by multiplying the length of one side by 6 (the number of sides in a hexagon):

Upper bound for perimeter = 6 × (3.6 + 0.05) = 21.9 cm (rounded to one decimal place)

2. Kelly's average speed can be calculated by dividing the distance she ran by the time she took:

Average speed = distance / time

The lower bound for the distance is 99.5 m (since 100 m was measured to the nearest meter, the actual distance could be as low as 99.5 m).

The lower bound for the time is 10.51 s (since 10.52 s was measured to the nearest hundredth of a second, the actual time could be as low as 10.51 s).

Therefore, the lower bound for Kelly's average speed is:

Average speed = 99.5 / 10.51 = 9.4617 m/s (rounded to 4 decimal places)

3. The length of the rectangle is 645 mm correct to the nearest 5 mm, which means it could be as small as 642.5 mm or as large as 647.5 mm. We can express this as:

645 mm ± 2.5 mm, similarly

400 mm ± 2.5 mm

Lower bound for length = 645 - 2.5 = 642.5 mm

Lower bound for width = 400 - 2.5 = 397.5 mm

Lower bound for area = 642.5 × 397.5 = 255393.75 mm²

Rounded to 3 significant figures, the lower bound for the area of the rectangle is 2.55 × 10⁵ mm².

4. To calculate the upper bound for the area of the rectangle, we need to multiply the upper bounds for the length and width of the rectangle:

Upper bound for length = 35 + 0.45 = 35.45 cm

Upper bound for width = 26 + 0.45 = 26.45 cm

Upper bound for area = 35.45 × 26.45 = 937.6525 cm²

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Using the given information, we found the angles of the right triangle as:

∠R = 56.30º

∠P = 33.7º

∠Q = 90º

What is a right triangle?

A triangle with a right angle, also known as a right-angled triangle or right-angled triangle, more technically an orthogonal triangle, has two perpendicular sides. The foundation of trigonometry is the relationship between the sides and various angles of the right triangle. A right triangle's hypotenuse is its longest side, its "opposite" side is the one that faces a certain angle, and its "adjacent" side is the one that faces the angle in question. To describe the sides of right triangles, we utilise specific terminology. The side opposite the right angle is always the hypotenuse of a right triangle.

Given one of the angles of the triangles as 90º.

So it is a right triangle.

PQ = 7.5cm
QR = 5cm
∠Q = 90º

The figure is given below.

Now PR is the hypotenuse.

Hypotenuse = [tex]\sqrt{base^2+height^2}[/tex]

base = 5 = QR

height = 7.5 = PQ

PR = [tex]\sqrt{5^2+7.5^2} = \sqrt{81.25}[/tex] = 9.01 cm

We can find the angles using trigonometric relations.

sin R = PQ/PR = 7.5/9.01 = 0.832

∠R = 56.30º

∠P = 180 - (∠Q+∠R) = 180 - (90+56.30) = 33.7º

Therefore using the given information, we found the angles of the right triangle as:

∠R = 56.30º

∠P = 33.7º

∠Q = 90º

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Answer: -7.6n + 11.1

Step-by-step explanation:

We will simplify the given expression by combining like-terms.

   Given:

3.1 - 4.6n - 3n + 8

  Subtract like-terms:

3.1 - 7.6n + 8

   Add like-terms:

11.1 - 7.6n

Answer:

[tex] \sf \: -7.6n + 11.1 [/tex]

Step-by-step explanation:

Now we have to,

→ Simplify the given expression.

The expression is,

→ 3.1 - 4.6n - 3n + 8

Let's simplify the expression,

→ 3.1 - 4.6n - 3n + 8

→ -4.6n - 3n + 3.1 + 8

→ (-4.6n - 3n) + (3.1 + 8)

→ (-7.6n) + (11.1)

→ -7.6n + 11.1

Hence, the answer is -7.6n + 11.1.

Answer:

It's very difficult

maby

Answer:

Population: The population is the total demand (sale) of the product over all days.

Variable of interest: The variable of interest is the daily demand (sale) of the product.

To obtain the confidence interval for the average daily demand at a confidence level of 97%, we can use the following formula:

Confidence interval = sample mean ± (t-value x standard error)

where t-value is the value from the t-distribution for the desired confidence level and degrees of freedom, and the standard error is calculated as:

standard error = sample standard deviation / √n

where n is the sample size.

Using the given data, we can calculate:

Sample mean = (35+44+38+55+33+56+60+45+48+40+45+35+42)/12 = 44.5

Sample standard deviation = 9.92

Degrees of freedom = n-1 = 12-1 = 11

From the t-distribution table with 11 degrees of freedom and a confidence level of 97%, the t-value is approximately 2.718.

Therefore, the confidence interval for the average daily demand is:

Confidence interval = 44.5 ± (2.718 x 9.92/√12) = 44.5 ± 9.14

The lower limit is 44.5 - 9.14 = 35.36 and the upper limit is 44.5 + 9.14 = 53.64.

So, we can say with 97% confidence that the true population average daily demand falls within the range of 35.36 to 53.64 thousand units.

If Aaron is 8 years older than Judi, then Aaron's age is 22 years , Judi's age is 14 years and Maree's age is 7 years .

Let Maree's age be = M;

Judi is twice as old as Maree, which means ⇒ Judi's age is 2M;

And Aaron is 8 years older than Judi, which means

⇒ Aaron's age is (2M+8);

We know that the sum of their ages is 43, so we can write an equation:

⇒ M + 2M + (2M + 8) = 43

Simplifying and solving for M:

We get,

⇒ 5M + 8 = 43

⇒ 5M = 35

⇒ M = 7

So, Maree's age is 7 years.

Now, we use this to find Judi's and Aaron's ages:

⇒ Judi = 2M = 2 × 7 = 14

⇒ Aaron = 2M + 8 = 2 × 7 + 8 = 22

Therefore, Judi is 14 years old and Aaron is 22 years old.

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Answer:

Step-by-step explanation:

To solve the equation 9x^2 - 36x = 0 by using the square root method, we first need to rearrange the terms to get x^2 and x on one side:

9x^2 - 36x = 0

Factor out 9x from the left-hand side:

9x(x - 4) = 0

Now we have two factors: 9x = 0 and x - 4 = 0. Solving for x in each factor gives us:

9x = 0: x = 0

x - 4 = 0: x = 4

Therefore, the solutions to the equation are x = 0 and x = 4.

Answer:

x = 0, x = 4

Step-by-step explanation:

Unfortunately, the equation 9x^2 - 36x = 0 cannot be solved using the square root method directly. The square root method is used to solve quadratic equations of the form ax^2 + bx + c = 0 by isolating the x^2 term, taking the square root of both sides, and solving for x. However, in the given equation, there is no constant term (c = 0), and therefore, we need to use a different method to solve it.

As I mentioned earlier, we can factor the equation and use the zero product property to solve for x. This method involves finding two factors of the quadratic equation that multiply to give 0, setting each factor equal to 0, and solving for x. In this case, we can factor out x and obtain the factors x and (9x - 36), which multiply to give 0. By setting each factor equal to 0 and solving for x, we obtain the solutions x = 0 and x = 4.

To solve the equation 9x^2 - 36x = 0 using the factorization method:

Factor out x from the left-hand side of the equation to get:

x(9x - 36) = 0

Apply the zero product property, which states that if the product of two factors is zero, then at least one of the factors must be zero. So, set each factor equal to zero and solve for x:

x = 0 or 9x - 36 = 0

For the second equation, solve for x:

9x - 36 = 0

9x = 36

x = 4

Therefore, the solutions to the equation are x = 0 and x = 4.

Note that this method involves factoring the quadratic equation and then using the zero product property to obtain the solutions. It works for any quadratic equation of the form ax^2 + bx + c = 0, where a, b, and c are constants, and a is not equal to zero.

The length of line JC is 20 miles.

What is Pythagorean theorem?

The Pythagorean theorem is a fundamental concept in geometry that states that in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of sides of triangle.

To find the length of line JC, which is the hypotenuse of the right triangle JSC, we can use the Pythagorean theorem, which states that in a right triangle, the sum of the squares of the two shorter sides (the legs) is equal to the square of the length of the longest side (the hypotenuse).

In this case, we have:

JS² + SC² = JC²

Substituting the given values, we get:

12² + 16² = JC²

144 + 256 = JC²

400 = JC²

Taking square root both sides, we get:

JC = √400JC = 20

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Answer: x=10/9

Step-by-step explanation:

around 13 jars of paint will the farmer need to buy in order to paint the entire exterior of the silo.

To find the number of jars the rancher required to get, you really want to know the surface area of both the cone and cylinder.

The method for finding the SA of the cone would be area =[tex]3.14 x r^2 + 3.14 x r x sqrt(r^2 + h^2)[/tex].

SA= 1976.0617791 [tex]m^2[/tex].

The method for finding the SA of the cylinder under the cone would be 3.14 x d x (d/2 + h).

A = 3097.6103564 [tex]m^2[/tex]

Then, at that point, you need to add 1976.0617791+3097.6103564 which gives you 5,073.8. (5,073 is the total surface area of the storehouse)

Then, the partition that by 400.

5,073.8/400 = 12.7

He would have to purchase around 13 jars of paint to have the option to paint the whole storehouse.

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ECDF=1/100

The value of the empirical cumulative distribution function (ECDF) at the smallest observation in a sample of 100 observations is 1/100.What is the empirical cumulative distribution function (ECDF)?The empirical cumulative distribution function (ECDF) is a graph that depicts the distribution of a given data set. It is used to illustrate the proportion or percentage of data points that fall below a particular value in the distribution. The ECDF can be used to construct a distribution function for data that do not have an existing theoretical distribution.The ECDF value at the smallest observation in a sample of 100 observations is 1/100. This is because the ECDF takes on a value of 0 for all values that are smaller than the smallest observation. Since there are 100 observations in the sample, each observation has a weight of 1/100. Thus, the ECDF value at the smallest observation is 1/100.

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Based on the difference between the temperature when Casey went to bed and when he woke up, the temperature when he went to bed was 7 degrees, which was 12 degrees warmer than -5 degrees.

The temperature difference is determined using subtraction.

Subtraction is one of the four basic mathematical operations, involving the minuend, the subtrahend, and the result of the operation called the difference.

The temperature when Casey woke up to prepare for school = -5

The difference between the temperature when Casey went to bed and when he woke up = 12 degrees warmer.

The temperature when Casey went to bed = 7 degrees (12 - 5)

Thus, we can conclude that Casey had a temperature of 7 degrees when he went to bed but woke up when it was -5 degrees.

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