Question 8 (06.02 LC)
Which statement best describes the expression 2y ÷ 7?
A. 2 times y divided by 7
B. 7 divided by 2 times y
C. 7 times y divided by 2
D. 2 divided by 7 times y

Question 9 (06.06 LC)
Which algebraic property is used to rewrite 2(x + 3) as 2x + 6?

A. Distributive property
B. Associative property
C. Commutative property of addition
D. Commutative property of multiplication

Question 10 (06.02 LC)
Which of these is the algebraic expression for "5 times the sum of 2 and y?"

A. 5 ⋅ 2 + y
B. 2 + 5 ⋅ y
C. 2(5 + y)
D. 5(2 + y)

Answer:

The distance between them in 2.5 min will be 320 ft

Step-by-step explanation:

Let

x--------------> Bill distance after 2.5  min------- > speed  45 ft/min

y--------------> Greg distance after 2.5  min------> speed 75 ft/min

z-------------> distance between them in 2.5 min

we Know that

z=x+y+20

Remember that

To find the distance multiply the speed by the time

so

x=45*2.5=112.5 ft

y=75*2.5=187.5 ft

therefore

z=x+y+20-----------> 112.5+187.5+20=320 ft

Final answer:

Bill and Greg's total distance apart after 2.5 minutes is 320 feet, calculated by adding their initial distance of 20 feet to the distance they covered in 2.5 minutes at their combined speed of 120 feet per minute.

Explanation:

Bill and Greg are walking away from each other with speeds of 45 and 75 feet per minute, respectively.

Since they are moving in opposite directions, their relative speed is the sum of their individual speeds, which is 45 + 75 feet per minute, or 120 feet per minute.

To find the distance between them after 2.5 minutes, we multiply their relative speed by the time elapsed. Therefore, the distance is:

Distance = Speed × Time = 120 feet/minute × 2.5 minutes = 300 feet.

We must also include the initial distance between them, which was 20 feet, so the total distance after 2.5 minutes is:

Total distance = Initial distance + Distance covered in 2.5 minutes = 20 feet + 300 feet = 320 feet.

Final answer:

When Isaac uses 24cm of string to make rectangles, no matter the individual dimensions, the sum of the length and width of these rectangles must always be 12 cm. This is a direct result of the fixed perimeter, which requires that the combined measurements of length and width halves the total perimeter to maintain the consistent total string usage.

Explanation:

Isaac made all these rectangles with 24cm lengths of string, which implies that the perimeter of all these rectangles is equal. The observation about the sum of the length and the width of these rectangles is that regardless of the individual measurements of the length and width, their sum must always be half the total perimeter of the rectangle. Given a fixed perimeter of 24cm, the formula for the perimeter of a rectangle (P) is

P = 2(l + w),

where l is the length and w is the width. Since the perimeter is given as 24cm, this simplifies to

24 = 2(l + w), or 12 = l + w.

This equation means that the sum of the length and width of the rectangles must always equal 12 cm. This is because the total loop made by the string when forming the rectangle has to be divided into four edges, two lengths, and two widths, which totals the provided 24cm of string. Therefore, if we adjust either the length or the width, the other measurement must adjust in such a way that their sum remains constant at 12 cm to maintain the total perimeter as 24cm.

About 15 tons of trout were caught in the lake in the years 1999 and 2011.

To determine the years when about 15 tons of trout were caught in the lake, we need to solve the equation [tex]\( y = -0.08x^2 + 1.6x + 10 \) for \( y = 15 \).[/tex]

The equation becomes:

[tex]\[ 15 = -0.08x^2 + 1.6x + 10 \][/tex]

Subtract 15 from both sides to set the equation to zero:

[tex]\[ 0 = -0.08x^2 + 1.6x - 5 \][/tex]

This is a quadratic equation in the form of [tex]\( ax^2 + bx + c = 0 \)[/tex], where:

- ( a = -0.08 )

- ( b = 1.6 )

- ( c = -5 )

We solve this quadratic equation using the quadratic formula [tex]\( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):[/tex]

First, calculate the discriminant [tex]\( \Delta = b^2 - 4ac \):[/tex]

[tex]\[ \Delta = (1.6)^2 - 4(-0.08)(-5) \][/tex]

[tex]\[ \Delta = 2.56 - 1.6 \][/tex]

[tex]\[ \Delta = 0.96 \][/tex]

Now, calculate the two possible values of ( x ):

[tex]\[ x = \frac{-1.6 \pm \sqrt{0.96}}{2(-0.08)} \][/tex]

Calculate [tex]\( \sqrt{0.96} \):[/tex]

[tex]\[ \sqrt{0.96} \approx 0.9798 \][/tex]

Now solve for ( x ):

[tex]\[ x_1 = \frac{-1.6 + 0.9798}{-0.16} \][/tex]

[tex]\[ x_1 = \frac{-0.6202}{-0.16} \][/tex]

[tex]\[ x_1 \approx 3.88 \][/tex]

[tex]\[ x_2 = \frac{-1.6 - 0.9798}{-0.16} \][/tex]

[tex]\[ x_2 = \frac{-2.5798}{-0.16} \][/tex]

[tex]\[ x_2 \approx 16.12 \][/tex]

Since ( x ) is the number of years since 1995:

For [tex]\( x_1 \approx 3.88 \):[/tex]

[tex]\[ \text{Year} = 1995 + 4 = 1999 \][/tex]

For [tex]\( x_2 \approx 16.12 \):[/tex]

[tex]\[ \text{Year} = 1995 + 16 = 2011 \][/tex]

Therefore, about 15 tons of trout were caught in the lake in the years 1999 and 2011.

The sperm whale's underwater duration is seven times that of the sea cow, showcasing remarkable breath-holding abilities, likely adapted for deep-sea foraging and hunting.

The relationship between the underwater durations of a sperm whale and a sea cow suggests that the sperm whale's capability is sevenfold that of the sea cow. Let's denote the sea cow's duration as "x." Accordingly, the sperm whale's underwater duration would be 7 times x.

In practical terms, if we assume the sea cow can stay submerged for 10 minutes (x = 10), then the sperm whale, being seven times more proficient, can stay underwater for 7 times 10, which is 70 minutes (7x = 7 * 10 = 70). This sevenfold difference implies a significant contrast in their breath-holding abilities.

The numerical representation extends universally; if the sea cow can endure underwater for "x" units of time, the sperm whale can endure for 7 times "x" units of time. This relationship showcases the remarkable adaptation of the sperm whale to extended submersion, likely linked to its deep-sea foraging behavior and hunting strategies.

In essence, the sperm whale can stay underwater for a duration that is seven times longer than that of the sea cow, highlighting its remarkable breath-holding capacity.

A sperm whale can stay underwater for 7 times longer than a sea cow, so it can stay underwater for 7 times the duration of a sea cow.

If a sperm whale can stay underwater 7 times longer than a sea cow, we can express this relationship as:

Time a sperm whale can stay underwater = 7 * Time a sea cow can stay underwater

So, if we let the time a sea cow can stay underwater be represented by 'x', then the time a sperm whale can stay underwater is:

7 * x

Therefore, a sperm whale can stay underwater for 7 times longer than a sea cow, which means it can stay underwater for 7x.

Answer:

(A) [tex]x=7.2 ft[/tex]

Step-by-step explanation:

Given: A triangle whose base angle is 44° and hypotenuse is 10ft.

To find: The value of x.

solution: Using the trigonometry, we have

[tex]\frac{x}{10}=cos44^{{\circ}}[/tex]

⇒[tex]x=10{\times}cos44^{{\circ}}[/tex]

⇒[tex]x=10{\times}0.72[/tex]

⇒[tex]x=7.2 ft[/tex]

Thus,the value of [tex]x=7.2 ft[/tex]

Answer: A. 7.2 ft

Step-by-step explanation:

Given: A right triangle with hypotenuse 10 ft and base angle [tex]44^{\circ}[/tex]

By using Trigonometry, we have

[tex]\cos (44^{\circ})=\frac{\text{side adjacent to }44^{\circ}}{\text{hypotenuse}}\\\\\Rightarrow \cos (44^{\circ})=\frac{x}{10}\\\\\Rightarrow0.71933980033=\frac{x}{10}\\\\\Rightarrow\ x=10\times0.71933980033=7.1933980033\approx7.2\ ft.[/tex]

Hence, the value of x = 7.2 ft.

Answer:

B is correct.

Step-by-step explanation:

Final answer:

The slope of the skateboard ramp, calculated using the rise over run formula, is 0.3. This is found by dividing the rise of 1.2 meters by the run of 4 meters.

Explanation:

The slope of a ramp is calculated as the rise divided by the run. To find the slope of the skateboard ramp, which rises 1.2 meters above the ground and runs 4 meters horizontally at the base, you would use the formula:

Slope (m) = Rise / Run

Here, the rise is the height the ramp rises above the ground, which is 1.2 meters, and the run is the distance along the ground, which is 4 meters horizontally. By substituting the given rise and run into the formula, you obtain:

Slope (m) = 1.2 m / 4 m = 0.3

So, the slope of the skateboard ramp is 0.3.

Answer: d.) The area of the new circle is 16 times larger.

Step-by-step explanation:

Let the radius of the original circle be r.

Then the area of the original circle will be :-

[tex]\text{A}_1=\pi r^2[/tex]

When she multiplies her original radius by four and repaints the circle.

The new radius = [tex]R=4r[/tex]

Then the area of the new circle will be :-

[tex]\text{Area}=\pi R^2\\\\\Rightarrow\ \text{Area}=\pi (4r)^2\\\\\Rightarrow\ \text{Area}=\pi (16r^2)\\\\\Rightarrow\ \text{Area}=16\pi r^2\\\\\Rightarrow\ \text{Area}=16(\pi r^2)\\\\\Rightarrow\ \text{Area}=16A_1[/tex]

Hence, The area of the new circle is 16 times larger.

Answer:

The correct answer is pilot. I took the test

Step-by-step explanation:

Answer:

The correct option is 2.

Step-by-step explanation:

Given information: WZ is tangent to circle O at point B.

According to the tangent of circle theorem: The tangent line is perpendicular to the radius at the point of tangency. It means the angle formed on the point of tangency by tangent and radius is a right angle.

Since WZ is tangent to circle O at point B and OB is radius, therefore

[tex]\angle OBZ=90^{\circ}[/tex]

Therefore correct option is 2.

The measure of angle ∠OBZ is 90°, as the tangent line to a circle at a point is perpendicular to the radius at that point.

The question regards the measure of angle ∠OBZ where WZ↔ is tangent to circle O at point B. By a well-known theorem in geometry, a line tangent to a circle is perpendicular to the radius drawn to the point of tangency. Therefore, the measure of angle ∠OBZ, which is formed by the radius OB and the tangent line at B, must be 90°.

Answer:

There are infinitely many solutions.

Step-by-step explanation:

While solving a system of equations, if we get a true statement like 1=1, then there will be "infinitely many solutions" and if we get a false statement like 1=2, then there will be "no solution".

Here the statement is 4=4, which is a true statement.

So, there are infinitely many solutions.

The possible perimeters for the squares are 20 cm and 12 cm, depending on how the wire is cut.

Let's denote the length of one part of the wire as x and the length of the other part as 32−x. Each part is bent to form a square.

The perimeter (P) of a square is given by 4 × side length.

The side length of the first square ([tex]S_1[/tex] ) is x/4, and the side length of the second square ([tex]S_2[/tex] ) is (32−x)/4.

The total area of the two squares is given as [tex]34cm^2[/tex] , so we can write the equation:

[tex]S^2_1 + S^2_2 = 34[/tex]

Substitute the expressions for [tex]S_1 and S_2[/tex] :

[tex](\frac{x}{4})^2 + (\frac{(32-x)}{4})^2 =34[/tex]

Now, solve for x:

[tex]\frac{x^2}{16} + \frac{(32-x)^2}{4} =34[/tex]

Multiply both sides by 16 to get rid of the denominators:

[tex]x^2 +(32-x)^2=544[/tex]

Expand and simplify:

[tex]x^2+1024-64x+x^2 =544[/tex]

Combine like terms:

−64x+480=0

Divide the entire equation by 2 to simplify:

−32x+240=0

Now, factor the quadratic equation:

(x−20)(x−12)=0

So, x=20 or x=12.

If x=20, the lengths of the two parts are 20 cm and

32−20=12 cm.

If x=12, the lengths of the two parts are 12 cm and

32−12=20 cm.

Now, we can find the perimeter of each square by multiplying the side length by 4:

For x=20, the perimeter is [tex]4 \times \frac{20}{4} = 20 cm[/tex]

For x=12, the perimeter is [tex]4 \times \frac{12}{4} = 12 cm[/tex]

So, the possible perimeters for the squares are 20 cm and 12 cm, depending on how the wire is cut.

Answer:

Robin is 9 years old and Batman is 36 years old

Step-by-step explanation:

Batman: 4x

Robin: x

4x + x = 45

5x = 45

x= 9

Answer:  the correct option is (D) [tex]x=\dfrac{k-11}{3}.[/tex]

Step-by-step explanation:  We are given to solve the following equation for the value of x :

[tex]3x+11=k~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(i)[/tex]

To solve the given equation for x, we need to take x on one side and all other terms on the other side of the equality.

The solution of equation (i) is as follows :

[tex]3x+11=k\\\\\Rightarrow 3x=k-11\\\\\Rightarrow x=\dfrac{k-11}{3}.[/tex]

Thus, the required value of x is [tex]x=\dfrac{k-11}{3}.[/tex]

Option (D) is CORRECT.

Answer: 56.25

Step-by-step explanation:

Since there are four sides to a square and the area is 225 yards you would -

225 yards/ 4 sides

= 56.25 yards per side

Answer:3,4.b

Step-by-step explanation: Took the test.