Which statement about the figure below is true?


Select one:
a. Points A, B, and E are not collinear.
b. Points C, F, B, and E are coplanar.
c. Points F, B, and C are collinear.
d. Points C, D, B, and F are coplanar.

We are given transformation rule

(x, y) --> (x + 4, y + 7).

The coordinate of the pre-image is (4,5).

The coordinate of the transformed image is (0,-2).

Please see, if we apply

(x, y) --> (x + 4, y + 7) rule.

(4,5) coordinate would become (4 +4 , 5+7 ) = (8,12).

But we need (0,-2) instead of (8,12).

Let us reverse operation in the given rule and check.

Let us change (x + 4, y + 7)  to (x-4, y-7) and now check.

(4,5) coordinate would become (4-4 , 5-7) = (0,-2).

So, we got exact coordinate of the transformed image.

Therefore, error was

(x, y) --> (x + 4, y + 7) rule should be (x, y) --> (x - 4, y - 7) rule.

Transformation involves changing the position of a point.

John's error is that, he subtracted the points instead of adding them.

The pre-image is given as:

[tex]\mathbf{(x,y) = (4,5)}[/tex]

The image is given as:

[tex]\mathbf{(x,y) = (0,-2)}[/tex]

The transformation rule is given as:

[tex]\mathbf{(x,y) \to (x + 4,y + 7)}[/tex]

When the rule [tex]\mathbf{(x,y) \to (x + 4,y + 7)}[/tex] is applied on [tex]\mathbf{(x,y) = (4,5)}[/tex], we have:

[tex]\mathbf{(x,y) = (4+4,5+7)}[/tex]

[tex]\mathbf{(x,y) = (8,12)}[/tex]

Hence, John's claim is incorrect.

His error is that, he subtracted the points instead of adding them.

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Final answer:

The lowest common denominator for the fractions 8/64 and 8/32 is 32. This is found by identifying 32 as the least common multiple of the denominators 64 and 32, allowing both fractions to be expressed with the same denominator.

Explanation:

Finding the Lowest Common Denominator

To find the lowest common denominator (LCD) for the fractions 8/64 and 8/32, you must identify the smallest number that both denominators can divide into without leaving a remainder. Both 64 and 32 can be divided by 32, so the LCD for these fractions is 32. It's important to understand that finding an LCD involves looking for the least common multiple (LCM) of the denominators. In this case, since 32 is a multiple of 64 (32x2=64), it serves as the LCM of 32 and 64, and consequently, the LCD of the fractions.

To further clarify, each fraction can be expressed with a denominator of 32. For 8/64, when we divide both the numerator and denominator by 8, we get 1/8. This fraction can then be converted to have a denominator of 32 through multiplication by 4 (1/8 * 4/4 = 4/32). Similarly, the fraction 8/32 is already expressed with the desired denominator.

By finding the LCD, it is easier to perform addition, subtraction, or comparison of fractions, as they will share the same denominator.

The lowest common denominator for the fractions [tex]\( \frac{8}{64} \) and \( \frac{8}{32} \) is \( \boxed{64} \).[/tex]

To find the lowest common denominator (LCD) for the fractions [tex]\( \frac{8}{64} \) and \( \frac{8}{32} \)[/tex], we need to determine the least common multiple (LCM) of the denominators 64 and 32.

Step 1: Find the prime factorization of each denominator:

[tex]- \( 64 = 2^6 \)[/tex]

[tex]- \( 32 = 2^5 \)[/tex]

Step 2: Determine the LCM by taking the highest power of each prime that appears in any factorization:

[tex]\[ \text{LCM} = 2^{\max(6, 5)} = 2^6 = 64 \][/tex]

Therefore, the LCD of the fractions [tex]\( \frac{8}{64} \) and \( \frac{8}{32} \) is \( 64 \).[/tex]

Step 3: Verify that ( 64 ) is indeed the LCD:

[tex]- \( \frac{8}{64} = \frac{1}{8} \)[/tex]

[tex]- \( \frac{8}{32} = \frac{1}{4} \)[/tex]

Both fractions can be rewritten with a denominator of ( 64 ):

[tex]- \( \frac{1}{8} = \frac{8}{64} \)[/tex]

[tex]- \( \frac{1}{4} = \frac{16}{64} \)[/tex]

Therefore, ( 64 ) is the lowest common denominator for[tex]\( \frac{8}{64} \) and \( \frac{8}{32} \).[/tex]

Thus, the lowest common denominator for the fractions [tex]\( \frac{8}{64} \) and \( \frac{8}{32} \) is \( \boxed{64} \).[/tex]

Answer:

C

Step-by-step explanation:

Answer:

The correct answer is option A,

One cm represented 11 in. in the first scale, but now 1 cm represents 12 in. in the second scale

Step-by-step explanation:

The earlier planned length was [tex]44 in\\[/tex]

So the scale of the map was

[tex]\frac{44}{4} \\= 11\\1 cm = 11 in\\[/tex]

The second planned length was [tex]48 in\\[/tex]

So the second scale of the map was

[tex]\frac{48}{4} \\= 12\\1 cm = 12 in\\[/tex]

Thus, option A is correct.

The answer is:

area of base =  [tex]12\sqrt{3}[/tex]

Hope it helps!

Probability is the chance of an event to occur from a total number of outcomes.

The formula for probability is given as:

Probability = Number of required events / Total number of outcomes.

The probability that a randomly selected game from the season was played by the wolves given that it was a loss is 1/5.

It is the chance of an event to occur from a total number of outcomes.

The formula for probability is given as:

Probability = Number of required events / Total number of outcomes.

We have,

Total number of wins and losses = 12 + 14 + 16 + 18 = 60

Number of losses wolves has = 12

Now,

The probability that a randomly selected game from the season was played by the wolves given that it was a loss.

= 12/60

= 1/5

Thus,

The probability that a randomly selected game from the season was played by the wolves given that it was a loss is 1/5.

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This is the angle measured in radians for an arc that is 1 kilometre long is [tex]\({\theta = \frac{360}{\pi^2}}\)[/tex] radians.

To find the angle measure of an arc in radians, we use the formula:

[tex]\[ \text{Angle in radians} = \frac{\text{Arc length}}{\text{Radius}} \][/tex]

Given that the diameter of the circle is 8 kilometres, we can find the radius by dividing the diameter by 2:

[tex]\[ \text{Radius} = \frac{\text{Diameter}}{2} = \frac{8 \text{ km}}{2} = 4 \text{ km} \][/tex]

The arc length is given as [tex]\(\pi\)[/tex] kilometres. Now we can calculate the angle in radians subtended by the arc:

[tex]\[ \theta = \frac{\text{Arc length}}{\text{Radius}} = \frac{\pi \text{ km}}{4 \text{ km}} = \frac{\pi}{4} \][/tex]

However, this is not the final answer. We need to find the angle measured in degrees. To convert radians to degrees, we use the conversion factor:[tex]\[ 1 \text{ radian} = \frac{180}{\pi} \text{ degrees} \][/tex]

Thus, the angle in degrees is:

[tex]\[ \theta_{\text{degrees}} = \theta_{\text{radians}} \times \frac{180}{\pi} = \frac{\pi}{4} \times \frac{180}{\pi} = \frac{180}{4} = 45 \text{ degrees} \][/tex]

Now, we need to find the angle measured in radians that correspond to an arc length of 1 kilometre. Using the same formula:

[tex]\[ \theta_{\text{1 km}} = \frac{1 \text{ km}}{4 \text{ km}} = \frac{1}{4} \text{ radians} \][/tex]

To find the number of radians in the entire circle, we multiply by [tex]\(2\pi\)[/tex](since the circumference is [tex]\(2\pi r\)[/tex] and the radius is 4 km):

[tex]\[ \text{Full circle in radians} = 2\pi \times 4 \text{ km} = 8\pi \text{ radians} \][/tex]

Now, we want to find how many times the arc length of 1 km fits into the full circle's circumference:

[tex]\[ \text{Number of arcs} = \frac{8\pi \text{ radians}}{1/4 \text{ radians}} = 32\pi \][/tex]

This represents the number of times the 1 km arc goes around the circle, which is also the measure of the angle in radians that corresponds to a 1 km arc. Since the full circle is [tex]\(360\)[/tex] degrees, the angle measured in degrees for the 1 km arc is:

[tex]\[ \theta_{\text{1 km degrees}} = 32\pi \times \frac{180}{\pi} = 32\pi \times \frac{360}{2\pi} = 32 \times 180 = 5760 \text{ degrees} \][/tex]

However, this is not the final answer. We made a mistake in the calculation. We should not multiply the number of arcs by the full circle's degrees. Instead, we should divide the full circle's degrees by the number of arcs to find the angle measure for one arc:

[tex]\[ \theta_{\text{1 km degrees}} = \frac{360}{32\pi} = \frac{360}{\pi} \times \frac{1}{32} = \frac{360}{\pi^2} \times \frac{\pi}{32} = \frac{360}{\pi^2} \text{ degrees} \][/tex]

To convert this back to radians, we multiply by [tex]\(\frac{\pi}{180}\)[/tex]:

[tex]\[ \theta_{\text{1 km radians}} = \frac{360}{\pi^2} \times \frac{\pi}{180} = \frac{360}{\pi^2} \text{ radians} \][/tex]

This is the angle measured in radians for an arc that is 1 kilometre long.

Through a time, distance and speed optimization problem, we find the least amount of time to reach the house is approximately 2.11 hours. The person should land the boat 2 miles from the house.

The problem given is a classic time, distance and speed optimization problem which can be solved with a bit of calculus and geometric reasoning. The first step is to define variables. Let's say the person decides to row to a point x miles down the shore, and then walk the remaining distance.

The time taken to row is the distance rowed divided by the rowing speed, and since the rowing distance is the hypotenuse of the right triangle formed, it is sqrt(4+[tex]x^2[/tex]) miles. Hence, time rowing is sqrt(4+[tex]x^2[/tex])/3 hours.

The distance walked is 6-x and hence, the time walking is (6-x)/5 hours.

The total time is then T(x)=sqrt(4+[tex]x^2[/tex])/3+(6-x)/5. The task is to find an x that minimizes T(x). To do so, take the derivative of T(x), set it equal to 0 and solve for x. After conducting these steps, you find that the minimum time is attained at x=2. Therefore, the least amount of time required would be T(2) which is approximately 2.11 hours. The person should land the rowboat 2 miles from the house.

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Final answer:

The function y = sin x + 1 is neither even nor odd because it does not satisfy the conditions f(x) = f(-x) for even functions and f(-x) = -f(x) for odd functions after substitution and simplification.

Explanation:

An even function is defined as a function that satisfies the condition f(x) = f(-x), showing symmetry about the y-axis. An odd function, on the other hand, satisfies the condition f(-x) = -f(x), indicating symmetry about the origin. To determine if y = sin x + 1 is an odd or even function, we must check these conditions.

Applying the even function definition, we would expect sin(-x) + 1 = sin(x) + 1 which is not true because sin(-x) = -sin(x) and thus does not satisfy the required symmetry. When we apply the odd function test, we expect -(sin x + 1) = sin(-x) + 1, which also doesn't hold because -sin(x) - 1 does not equal sin(-x) + 1. Therefore, y = sin x + 1 is neither odd nor even.

General Idea:

When we are working with word problems, we need to follow the below steps:

Step 1: Assign variable for the unknown that we need to find.

Step 2: Write a meaningful mathematical equation using the sentence given

Step 3: Solve the equation by Performing reverse operation by Undoing whatever is done to the variable. Solving means find the value of the variable which will make the equation TRUE.

Applying the concept:

Step 1: Let 'x' be the length of longest side of the fence.

Step 2: We need to set up an equation based on the information given.

[tex] Perimeter\; of\; rectangle=\; 2(\; Longest \; side\; +\; Shortest \; side\; ) [/tex]

Substituting 20 for the perimeter of rectangle, x for Longest side and [tex] \frac{7}{2} [/tex] for the shortest side in the above formula, we get the below equation.

[tex] 2(x+\frac{7}{2} )=20 [/tex]

Step 3: Solving the equation.

[tex] 2(x+\frac{7}{2} )=20\\ Distribute \; 2 \; in \; the \; left \; side \; of \; the \; equation\\ \\ 2x+2 \cdot \frac{7}{2} =20\\ Simplify \; in \; the\; left\; side \; of \;the \;equation\\ \\ 2x+7=20\\ Subtract \; 7 \;on\; both \;sides \; of\; the \; equation\\ \\ 2x+7-7=20-7\\ Combine\; like \; terms\\ \\ 2x=13\\ Divide \; by \; 2\;on \; both\; sides\\ \\ \frac{2x}{2} =\frac{13}{2} \\ Simplify \; fraction\;on \; both \; sides\\ \\ x=6.5 [/tex]

Conclusion:

The length of longest side of the fence is 6.5 feet.

Final answer:

The ratio of surface area to volume for the square prism is 4/9 when b = 12 ft and h = 18 ft.

Explanation:

The expression for the ratio of surface area to volume of the square prism is 2b + 4h/bh. To find the ratio when b = 12 ft and h = 18 ft, substitute these values into the expression.

Ratio = 2(12) + 4(18)/(12)(18)

= 24 + 72/216

= 96/216

= 4/9

Therefore, the ratio of surface area to volume for the given square prism is 4/9 when b = 12 ft and h = 18 ft.

The ratio of surface area to volume for a square prism with a base of 12 ft and a height of 18 ft is calculated using the formula 2b + 4h / bh, which results in a ratio of 4 / 9 ft⁻¹.

The student has asked to find the ratio of surface area to volume for a square prism when given specific dimensions. The formula provided is 2b + 4h / bh, where b is the base length and h is the height of the prism. To find the ratio for b = 12 ft and h = 18 ft, we substitute these values into the formula:

Ratio = (2 × 12 ft) + (4 × 18 ft) / (12 ft × 18 ft) =
24 ft + 72 ft / 216 ft² =
96 ft / 216 ft².

After performing the calculations, we simplify the expression to get the ratio:

Ratio = 96 ft / 216 ft² = 4 / 9 ft⁻¹.

Thus, the ratio of surface area to volume for a square prism with a base of 12 ft and a height of 18 ft is 4 / 9 ft⁻¹.

Answer:

Jon eats 2 apples; Jon eats more than 2 apples.

Step-by-step explanation:

Mutually exclusive events are events which have no common element between them.  They are completely disjoint and the intersection would be a null set

Hence probability for the intersection of mutually exclusive events =0

Here we are given 4 options to select.

Jon eats more than 1 apple; Jon eats 3 apples.

These two are not mutually exclusive as eating 3 apples includes eating 1 apple.

Jon eats 4 apples; Jon eats 1 apple.

These two are not mutually exclusive as eating 4 apples includes eating 1 apple.

Jon eats 2 apples; Jon eats more than 2 apples.

These two are mutually exclusive since he cannot eat 2 applies exactly and also more than 2 apples

Jon eats 2 apples; Jon eats 4 apples.

These  two are not mutually exclusive as eating 4 apples includes eating 2 apple.

Hence correct answer is

Jon eats 2 apples; Jon eats more than 2 apples.

Answer:

The correct option is D.

Step-by-step explanation:

In triangle △JKL,

[tex]\frac{JK}{KL}=\frac{30}{50}=\frac{3}{5}[/tex]

In triangle △MKN,

[tex]\frac{MK}{KN}=\frac{15}{25}=\frac{3}{5}[/tex]

In triangle △JKL and △MKN

[tex]\frac{JK}{KL}=\frac{MK}{KN}[/tex]

[tex]\angle JKL=\angle MKN[/tex]                       (Vertically opposite angles)

Since two sides are proportional and an inclined angle is congruent, so by SAS theorem of similarity we get

[tex]\triangle JKL=\triangle MKN[/tex]

Therefore option D is correct.

Answer:

The total number of pounds of cheese Kala bought is 1.75 pounds

Step-by-step explanation:

Kala bought two types of cheese at a deli.

She bought 0.50 pound of American cheese and 1.25 pounds of Swiss cheese.

The total number of pounds of cheese Kala bought

= 0.50+1.25

= 1.75 pounds

Hence,  the total number of pounds of cheese Kala bought is:

  1.75 pounds