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P. 6 Compare and order rational numbers: word problems ETK


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Manuel and his friends built model cars using pieces of wood and plastic wheels. They rolled


the cars down a ramp and measured to see whose car would coast the farthest. Manuel's car


coasted 10 feet, Richard's car coasted 10. 5 feet, and Diego's car coasted 10


2


feet.


6


How many of the cars coasted more than 10. 75 feet?


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-43x < 79 - 16

To solve the inequality −43x + 16 < 79, we need to isolate the variable x.

Let's begin by subtracting 16 from both sides of the inequality:

−43x + 16 - 16 < 79 - 16

Simplifying the equation, we have:

−43x < 63

Next, we divide both sides of the inequality by -43. However, when we divide by a negative number, the direction of the inequality sign will be flipped:

x > 63 / -43

Simplifying further, we have:

x > -1.465

Therefore, the solution to the inequality is x > -1.465.

In interval notation, we can represent the solution as (-1.465, ∞), indicating that x is greater than -1.465 and extends indefinitely towards positive infinity.

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A) The equation relates the number of cups of flour to the number of pound cakes that the bakery makes is f = 2.75p

B) Cups of flour are needed for 14 cakes is 38.5

A) The number of cups of flour, f, that a bakery needs for the number of pound cakes that they make, p is directly proportional to each other which can be written in form ,

f/p = 8.25/3

f = (8.25/3)×p

f = 2.75 p

The equation forms is f = 2.75p

B) Cups of flour are needed for 14 cakes

Here p = 14

by putting the value in the equation we get ,

f = 2.75(14)

f = 38.5

hence , cups of flour are needed for 14 cakes is 38.5

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Answer 1 inch per hour

Step-by-step explanation:

Answer:

1 inch per hour

Step-by-step explanation:

Let x represent inches burned per hour

The equation to finding the rate of the candle burning:

3x=3

For those who dont know, whenever you have a variable with a term, its basically multiplying, and we want to do the opposite to get x by itself, so we divide

x=3/3

Which can be simplified as:

x=1

Where x is the inches burned per hour, which equal 1

The height of the ice block must be 10 inches, the length and width could be any combination of dimensions that multiply together to equal 48 square inches.

The overall number of cube units that the cube totally occupies is the definition of a cube's volume. Volume is simply the total amount of space an object takes up. The cube's volume can be calculated using the formula a3 where an is the cube's edge.

given,

To check if the height of the ice block must be 10 inches to have a volume of 480 cubic inches, we can use the formula for the volume of a rectangular solid:

V = l * w * h

where l represents the length, w represents the measurement of width, while h is the peak, and V is the volume.

Since the volume is given as 480 cubic inches, and the height is specified as 10 inches, we can write:

480 = l * w * 10

Dividing both sides by 10, we get:

48 = l * w

This means that the product of the length and width must be equal to 48 square inches in order for the block of ice to have a volume of 480 cubic inches with a height of 10 inches.

There are many possible dimensions that satisfy this condition. For example, the block of ice could have dimensions of 8 inches by 6 inches by 10 inches, or 12 inches by 4 inches by 10 inches, or 16 inches by 3 inches by 10 inches, and so on.

Therefore, while the height of the ice block must be 10 inches, the length and width could be any combination of dimensions that multiply together to equal 48 square inches.

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Translation, rotation, and reflection, By following these transformations in sequence, you can map triangle ABC to triangle A'B'C'.

To map triangle ABC to triangle A'B'C', you would need to follow a specific sequence of transformations, which may include translation, rotation, and reflection. Here's a step-by-step explanation:

Step 1: Translation
Translate triangle ABC by a specific vector (x, y) so that point A moves to point A'. The same vector will also move points B and C to their corresponding new positions.

Step 2: Rotation
If triangle A'B'C' is rotated compared to the translated triangle, rotate the translated triangle around point A' by a specific angle, either clockwise or counterclockwise, until point B aligns with point B'.

Step 3: Reflection
If triangle A'B'C' is a mirror image of the rotated triangle, reflect the rotated triangle across a line of symmetry (usually a line passing through A'). This will change the orientation of the triangle and align point C with point C'.

By following these transformations in sequence, you can map triangle ABC to triangle A'B'C'. Keep in mind that the specific details of translation, rotation, and reflection will depend on the coordinates and orientation of the given triangles.

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The diameter of the circle whose circumference equals 172.7 cm is approximately 55 cm, as found using the formula c = πd with the given value of π as 3.14. The answer is option A.

The formula for the circumference of a circle is c = πd, where c is the circumference and d is the diameter. We are given that the circumference of the circle is 172.7 cm, and π is approximately 3.14. So we can solve for the diameter as

d = c/π = 172.7/3.14 ≈ 55

Therefore, the diameter of the circle is approximately 55 cm.

The correct answer is option A: 55 cm.

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If each locker is identified by one letter and one digit, the total number of lockers built in the school is 229.

There are 26 letters in the alphabet (A to Z), and if each locker is identified by one letter and one digit, then there are a total of 26 × 10 = 260 possible locker identifications using the letters A to Y and digits 0 to 9.

Out of these, there are 4 digits used to identify the lockers with the letter Z. So, there are 10 − 1 = 9 digits left to use for the remaining lockers, as we cannot use the digit already used for the lockers with the letter Z.

Similarly, there are 25 letters left to use for the remaining lockers, as we cannot use the letter Z.

Therefore, the number of remaining lockers is 9 × 25 = 225. Adding the 4 lockers with the letter Z, the total number of lockers built in the school is 225 + 4 = 229.

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Using cosine rule, the largest angle In the triangle is

The angle is found first using cosine rule by the formula

cos A = (b² + c² - a²) ÷ 2bc

Solving for the largest angle

substituting the values

cos γ = (2² + 1.5² - 3²) ÷ 2 * 1.5 * 2

< γ = arc cos ( -0.4583 )

< γ = 117.28 degrees

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Based on these calculations, option 1 gives you the cheaper price per square inch of pizza.

To determine which option will give you a cheaper price per square inch of pizza, you will need to compare the prices of the two options and calculate their respective prices per square inch.

Let's say option 1 is a large pizza with a diameter of 18 inches and costs $20, while option 2 is a medium pizza with a diameter of 14 inches and costs $15.

To calculate the area of each pizza, you need to use the formula for the area of a circle:

A = πr^2

For option 1, the radius is 9 inches (half of the diameter), so the area is:

A = π(9)^2 = 81π square inches

For option 2, the radius is 7 inches (half of the diameter), so the area is:

A = π(7)^2 = 49π square inches

Now you can calculate the price per square inch for each option by dividing the cost by the area:

Option 1:

Price per square inch = $20 / (81π) = $0.078/square inch

Option 2:

Price per square inch = $15 / (49π) = $0.098/square inch

Based on these calculations, option 1 gives you the cheaper price per square inch of pizza.

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The area of the rectangular park is equal to 0.233 sq miles.

The measurements of the park that are given in the question are given as 7/10 mile by 2/6 mile.

The length of the rectangle park is 7/10 and the width of the park is 2/6 mile. We know that the area of the rectangle park is given as the:

= length * width of the park.

= L * W

= (7/10) * (2/6)

we can reduce the fraction even further to make the calculation easy

= (7/10) * (1/3)

Multiplying the denominators we get

= 7/30

To make the answer even simpler it can be converted into a decimal form which will be:

= 0.233 sq miles.

Therefore, The area of the rectangular park is equal to 0.233 sq miles.

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The original number was 17 to which when Angad adds 20, then doubles it and gets an answer of 53.

To solve this problem, we need to work backwards from the final answer. If Angad added 20 to the original number, we can subtract 20 from 53 to get 33. This means that after Angad doubled the number, he had 33.

To find the original number, we need to undo the doubling process. If Angad had 33 after doubling the number, we can divide 33 by 2 to get the original number.

33 divided by 2 equals 16.5. However, we know that the original number must be a whole number since we cannot add 20 to a decimal. Therefore, we can round up to the nearest whole number to get 17.

So, the original number was 17.

In summary, to find the original number, we worked backwards from the final answer and undid the steps that Angad took. We subtracted 20 to undo the addition, then divided by 2 to undo the doubling. We rounded up to the nearest whole number since the original number must be a whole number.

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Answer:

Step-by-step explanation:

We have evidence to suggest that the true percentage of college degrees in mathematics is different from 1%.

The null hypothesis is a type of hypothesis that explains the population parameter and is used to examine if the provided experimental data are reliable.

To test the claim that the percentage of college degrees in mathematics is not 1%, we can use a hypothesis test. Let's assume the null hypothesis is that the true percentage of college degrees in mathematics is 1%, and the alternative hypothesis is that it is different from 1%.

- Null hypothesis: The percentage of college degrees in mathematics is 1%.

- Alternative hypothesis: The percentage of college degrees in mathematics is different from 1%.

We can use a binomial distribution to model the number of graduates with math degrees in a sample of 12,317. Under the null hypothesis, the expected number of graduates with math degrees is:

Expected value = sample size * probability of math degrees = 12,317 * 0.01 = 123.17

Since we are testing at a 0.10 level of significance, the critical values for a two-tailed test are ±1.645 (using a standard normal distribution table).

The test statistic can be calculated as:

z = (observed value - expected value) / standard deviation

The standard deviation of the binomial distribution can be calculated as:

√(sample size * probability of success * (1 - probability of success))

So,

standard deviation = √(123.17 * 0.01 * 0.99) = 1.109

The observed value is 148.

The test statistic is:

z = (148 - 123.17) / 1.109 = 22.38

Since the absolute value of the test statistic is greater than 1.645, we can reject the null hypothesis at the 0.10 level of significance.

Therefore, we have evidence to suggest that the true percentage of college degrees in mathematics is different from 1%.

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Answer: The given polynomial is -2x³.

a) To write the polynomial in standard form, we arrange the terms in descending order of degrees:

-2x³

b) The degree of a polynomial is the highest exponent of the variable in the polynomial. In this case, the degree of the polynomial is 3.

c) The lead coefficient is the coefficient of the term with the highest degree. In this case, the lead coefficient is -2.

Step-by-step explanation:

It seems that the question provided is not clear and has some typos or formatting issues, making it difficult to understand the exact problem you need help with. Please rephrase or clarify the question, and I'll be more than happy to help you!

To find W,(1,0), we need to use the formula for the partial derivative of F with respect to u at (6,7) and (6,-7) and plug in the given values:

F_u(6,7) = 6,0(6,7) = -5
F_u(6,-7) = 6,0(6,-7) = -7

Now we can use these values to solve for W,(1,0) using the formula:

W,(1,0) = F(6,t) - F_u(6,7)(1-6) - F_u(6,-7)(1-6)

Plugging in the given values, we get:

W,(1,0) = F(6,t) - (-5)(-5) - (-7)(-5)
W,(1,0) = F(6,t) + 30

We still need to find F(6,t). To do this, we use the formula for the partial derivative of F with respect to s at (1,0) and plug in the given values:

F_s(1,0) = 1,0(6,0) - 7,0(1,0) - 9,0(1,0)
F_s(1,0) = -7

Now we can use F_u(6,7), F_u(6,-7), and F_s(1,0) to solve for F(6,t) using the formula:

F(6,t) = F_u(6,7)(6,t-7) + F_u(6,-7)(6,t+7) + F_s(1,0)(t)

Plugging in the given values, we get:

F(6,t) = (-5)(6,t-7) + (-7)(6,t+7) + (-7)(t)
F(6,t) = -77t - 188

Now we can substitute this value of F(6,t) into our formula for W,(1,0) to get the final answer:

W,(1,0) = -77t - 188 + 30
W,(1,0) = -77t - 158

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Answer:

20

Step-by-step explanation:

(x-3)+(8x+3)=180

9x=180

x=20

The displacement of the man from Noth east of A from C is 12.2 km.

The displacement of the man is the distance between his initial position at A and final position at C.

The angle formed at position B is calculated as follows;

θ = 30⁰  + 90⁰

θ = 120⁰

The displacement of the man is calculated by applying cosine rule as follows;

d² = 8² + 6² - 2(8 x 6) cos (120)

d² = 148

d = √148

d = 12.2 km

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61.93% of the surveyed Maryland households purchase at least one prepared meal per week from the grocery store.

A national grocery chain is considering expanding their selection of prepared meals, and they believe that 67 percent of households purchase at least one meal per week from the grocery store. In a survey conducted in Maryland, 641 out of 1,035 households purchase at least one meal per week at the store.
To determine the percentage of Maryland households purchasing at least one prepared meal per week, follow these steps:
Divide the number of households purchasing at least one meal per week (641) by the total number of households surveyed (1,035).
Multiply the result by 100 to get the percentage.
Here's the calculation: (641 / 1,035) x 100 = 61.93%
So, 61.93% of the surveyed Maryland households purchase at least one prepared meal per week from the grocery store.

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The antiderivative of e^(kx), where k is any constant, is (1/k)e^(kx) + C, where C is the constant of integration.

1. As we can see, you are asked to find the antiderivative of the function e^(kx).
2. Recall that the antiderivative of a function is the function that, when differentiated, gives you the original function.
3. The derivative of the function (1/k)e^(kx) is e^(kx), as the constant k in the exponent gets multiplied by the (1/k) factor, canceling each other out.
4. So, the antiderivative of e^(kx) is (1/k)e^(kx) + C, where C is the constant of integration.

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The volume of a rectangular prism is calculated by multiplying the length, width, and height. In this case, we are given the length and width, but not the height. So, we cannot calculate the exact volume without knowing the height.

To find the volume of a rectangular prism, we need to multiply its length, width, and height.

Given:

Length = 2 units

Width = 7 1/3 units

To calculate the volume, we first need to convert the mixed fraction to an improper fraction.

7 1/3 = (7 * 3 + 1) / 3 = 22/3 units.

Now, we can calculate the volume:

Volume = Length * Width * Height

= 2 units * (22/3 units) * Height.

Since the height is not provided, we cannot calculate the exact volume without that information. However, if you provide the height of the rectangular prism, I can help you find the volume by substituting the value into the formula.

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If Jesse can mow 3 yards in 8 hours. Jackson can mow twice as many yards per hour the constant of proportionality between the number of yards Jackson can mow and the number of hours is 3/4.

To find the constant of proportionality between the number of yards Jackson can mow and the number of hours, we can use the formula:

k = y/x

where k is the constant of proportionality, y is the number of yards, and x is the number of hours.

We know that Jesse can mow 3 yards in 8 hours, which means his rate of mowing is: 3 yards/8 hours = 3/8 yards per hour

We also know that Jackson can mow twice as many yards per hour as Jesse, which means his rate of mowing is:

2 * (3/8) yards per hour = 3/4 yards per hour

Now we can use the formula to find the constant of proportionality for Jackson:

k = y/x = (3/4) yards per hour / 1 hour = 3/4

Therefore, the constant of proportionality between the number of yards Jackson can mow and the number of hours is 3/4.

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The velocity with which a ball reaches the ground when it is dropped

from a height of 64 m is 35.44m/sec

From the information given, we have that the equation representing the velocity of the ball is expressed as;

V = 4.43√h

Given that the parameters of the formula are;

Since the height of the ball from the ground is 64m, we have to substitute the value, we have;

V = 4.43√64

Find the square root of the value

V= 4.43(8)

Now, multiply both the values to determine the velocity, we get;

V = 35.44m/sec

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To determine whether the series น k2 sk converges, we can use the Integral Test. Let f(x) = x2, then f'(x) = 2x. Since 2x is continuous, positive, and decreasing on [1,∞), In summary, the series Σ (1/k^2) converges, while the series Σ (2 + (-1)^{5k}) does not converge.

∫1∞ f(x) dx = ∫1∞ x2 dx = lim (t → ∞) [1/3 x3]1t = ∞
Since the integral diverges, the series น k2 sk also diverges.
b. To determine whether the series 2+(-1){ 5k converges, we can use the Alternating Series Test. The series has alternating signs and the absolute value of each term decreases as k increases. Let ak = 2+(-1){ 5k, then:
|ak| = 2+1/32k ≤ 2
Also, lim (k → ∞) ak = 0. Therefore, by the Alternating Series Test, the series 2+(-1){ 5k converges.

a. For the series Σ (1/k^2) (denoted as น k2 sk), we can use the p-series test. A p-series is a series of the form Σ (1/k^p), where p is a constant. If p > 1, the series converges, and if p ≤ 1, the series diverges. In this case, p = 2, which is greater than 1. Therefore, the series Σ (1/k^2) converges.
b. For the series Σ (2 + (-1)^{5k}), we can use the alternating series test. An alternating series is a series that alternates between positive and negative terms. In this case, the series alternates because of the (-1)^{5k} term. However, the series does not converge to zero as k goes to infinity, since there is a constant term 2. Therefore, the series Σ (2 + (-1)^{5k}) does not converge.
In summary, the series Σ (1/k^2) converges, while the series Σ (2 + (-1)^{5k}) does not converge.

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The magnitude and direction of the vector u = <-4, 7> are |u| ≈ √65 and θ ≈ 119.74°.

To find the magnitude and direction of the vector u = <-4, 7>, we will use the following steps:

1. Calculate the magnitude using the Pythagorean theorem.
2. Calculate the direction using the arctangent function.

Step 1: Calculate the magnitude.
Magnitude (|u|) = √((-4)^2 + (7)^2) = √(16 + 49) = √65

Step 2: Calculate the direction (angle θ).
θ = arctan(opposite/adjacent) = arctan(7/-4) ≈ -60.26° (in degrees)
Since the vector is in the second quadrant, we need to add 180°.
θ = -60.26° + 180° ≈ 119.74°

So, the magnitude and direction of the vector u = <-4, 7> are |u| ≈ √65 and θ ≈ 119.74°.

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The horizontal distance between (16, -24) and (-3, -24) is 19 units.

Explanation:

We can calculate the horizontal distance by finding the difference between the x-coordinates of the two points.

Horizontal distance = difference in x-coordinates = (-3) - 16 = -19

However, the distance is negative, which doesn't make sense as distance is always positive. So we take the absolute value of the difference to get the actual distance.

|Horizontal distance| = |-19| = 19

Therefore, the horizontal distance between the two points is 19 units.

The first derivative of the implicit function given by x^(2/3) + y^(2/3) = 14 can be found using implicit differentiation. We take the derivative of both sides with respect to x and use the chain rule to differentiate the terms involving y:(2/3)x^(-1/3) + (2/3)y^(-1/3) * dy/dx = 0Then, we solve for dy/dx:dy/dx = -(x/y)^(1/3)This is the first derivative of the implicit function. To evaluate it at a specific point, we need to substitute the coordinates of that point into the equation above.

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The continuous compound rate of decay of this isotope is approximately 1.41%.

In order to find the continuous compound rate of decay of the radioactive isotope, we can use the formula:

N(t) = N0 * e^(-λt)

Where N(t) is the amount of the isotope present at time t, N0 is the initial amount of the isotope, λ is the continuous decay constant, and t is the time elapsed (in this case, 6 years).

We are given that 91.9% of the isotope is still present after 6 years, so:

0.919 = e^(-λ * 6)

To solve for λ, we can take the natural logarithm of both sides:

ln(0.919) = -6λ

Now, we can solve for the decay constant:

λ = -ln(0.919) / 6 ≈ 0.0141

So, the continuous compound rate of decay of this isotope is approximately 1.41%.

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By  null hypothesis there is sufficient evidence at the 5% significance level to purchase the proposed panels.

To determine if there is sufficient evidence to purchase the proposed panels, we can perform a hypothesis test. Let's define the following:

- p: the proportion of all panels that produce an efficiency rating greater than 20%

- p0: the proportion specified in the proposal, which is p0 = 0.18

- n: the sample size, which is n = 140

- x: the number of panels in the sample that produce an efficiency rating greater than 20%, which is x = 32

We want to test the null hypothesis H0: p <= p0 against the alternative hypothesis Ha: p > p0. The significance level is alpha = 0.05.

We can use the normal approximation to the binomial distribution since both np0 and n(1-p0) are greater than 10, where np0 is the expected number of panels that produce an efficiency rating greater than 20% under the null hypothesis.

Under the null hypothesis, the test statistic z is approximately:

z = (x - np0) / sqrt(np0(1-p0))

Plugging in the values, we get:

z = (32 - 140*0.18) / sqrt(140*0.18*0.82) ≈ 1.96

The critical value of z at alpha = 0.05 with a one-tailed test is 1.645. Since our calculated z value is greater than the critical value, we reject the null hypothesis.

Therefore, there is sufficient evidence at the 5% significance level to purchase the proposed panels.

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The finished table contains the following amount of cans per day:

To find out how many cans of food per day to give a cat, divide the cat's weight by 4. If the weight is not a multiple of 4, the result will be a fraction, which represents a fraction of a can of food.

A 5-pound cat needs 5/4 or 1.25 cans of food per day. Simplify this fraction to 1 ¹/₄ or 1.25.

Others include:

6/4 = 1.5

7/4 = 1.75

8/4 = 2

9/4 = 2.25

10/4 = 2.5

11/4 = 2.75

12/4 = 3

13/4 = 3.25

14/4 = 3.5

15/4 = 3.75

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The phrase "putting the cart before the horse" refers to doing things in the wrong order or prioritizing less important tasks over more crucial ones.

In the context of the puzzle time question, the answer is: You should never put the cart before the horse, as it is essential to follow the proper sequence of actions to achieve the desired outcome effectively.

In the context of the puzzle time question, the phrase "putting the cart before the horse" implies that it is crucial to follow the proper sequence of actions or steps to solve the puzzle correctly.

By prioritizing less important tasks or skipping essential steps, the desired outcome may not be achieved, and the solution may be flawed.

To effectively solve the puzzle, it is essential to understand and follow the rules, guidelines, and necessary steps in the correct order. This ensures that each action builds upon the previous one and leads to a coherent solution.

By doing so, you avoid the mistake of putting the cart (less important tasks) before the horse (more crucial steps), leading to a successful and accurate solution.

In summary, the phrase "putting the cart before the horse" emphasizes the importance of following the correct sequence of actions or steps in any given task, including solving puzzles. It serves as a reminder to prioritize tasks appropriately to achieve the desired outcome effectively.

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It would take Annmarie and Gladys 96 minutes to plow the field together.

Annmarie can plow a field in 240 minutes. Gladys can plow the same field 80 minutes faster than Annmarie.

So Gladys can plow the field in 240 - 80 = 160 minutes.

Let x be the time it takes for both of them to plow the field together.

The combined rate of Annmarie and Gladys is the sum of their individual rates.

Annmarie's rate is 1 field per 240 minutes, which is 1/240 field per minute.

Gladys's rate is 1 field per 160 minutes, which is 1/160 field per minute.

Their combined rate is:

1/240 + 1/160 = 1/x

Simplifying this equation:

1/x = (4/960) + (6/960) = 10/960

1/x = 1/96

Multiplying both sides by 96, we get:

x = 96 minutes

Therefore, it would take Annmarie and Gladys 96 minutes to plow the field together.

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