Mathswatch Question:


Liam is a tyre fitter.


It takes him 124 minutes to fit 4 tyres to a lorry.


a) How long would it take him to fit 6 tyres to a lorry. ?


b) If he works for 93 minutes, how many tyres can he fit?


Working out for question a:


a) 124×6÷4=186(minutes)


Correct answer for question a is 186.


Correct answer for question b is 3

Answer:

(1, 1.5).

Step-by-step explanation:

Midpoint of (x1, y1) and (x2, y2)

(x1 + x2)/2 , (y1 + y2)/2

=(4+-2)/2, (-2+5)/2

= (1, 1.5).

Answer:

1/11880 or 0.00008417508

Step-by-step explanation:

The probability of this can be determined by 1/12 x 1/11 x 1/10 x 1/9

We subtract one from the denominator each time because that ride has already been used, and cannot appear again in the list.

a. The tube dilution in tubes 1, 3, and 5:
- Tube 1: 1:5 (1 ml serum + 4 ml saline)
- Tube 3: 1:125 (1:5 dilution from Tube 1 x 1:5 dilution from Tube 2 x 1:5 dilution from Tube 3)
- Tube 5: 1:3125 (1:125 dilution from Tube 3 x 1:5 dilution from Tube 4 x 1:5 dilution from Tube 5)

b. The solution dilution in tubes 1, 2, and 7:
- Tube 1: 1:5
- Tube 2: 1:25 (1:5 dilution from Tube 1 x 1:5 dilution from Tube 2)
- Tube 7: 1:78125 (1:3125 dilution from Tube 5 x 1:5 dilutions for Tubes 6 and 7)

c. The total volume and solution dilution in tube 10 before transfer:
- Total volume: 3 ml (2 ml saline + 1 ml transferred from Tube 9)
- Solution dilution: 1:1953125 (1:78125 dilution from Tube 7 x 1:5 dilutions for Tubes 8, 9, and 10)

d. The amount or volume of serum in tube 6 before transfer and after transfer:
- Before transfer: 0.00064 ml (2 ml x 1:3125 dilution from Tube 5)
- After transfer: 0.00032 ml (1 ml x 1:3125 dilution from Tube 5, as half the volume was transferred to Tube 7)

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The Agnews' annual expenses are $12,205 less than their disposable income.

The amount of money a person or family has available to spend or save after paying taxes and other necessary costs like rent or mortgage payments, utilities, and insurance premiums is known as disposable income.

It stands for the money that is left over after taxes for discretionary expenses, such as savings or hobbies or amusement.

The Agnews' annual expenses are $39,826, and their disposable income is $52,031. To find out how much less their annual expenses are than their disposable income, we can subtract their annual expenses from their disposable income:

$52,031 - $39,826 = $12,205

Therefore, the Agnews' annual expenses are $12,205 less than their disposable income.

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Answer:

7 units

Step-by-step explanation:

Since DEF is a right triangle, and angle F is a right angle, DE is the hypotenuse, in which we can use a^2 + b^2 = c^2 25 to the power of 2 is 625 and 24 to the power of 2 is 576. 625-576 = 49. The square root of 49 is 7

The amount that Benjamin must save every month to pay off the discounted premium is $ 40. 80

The total premium for the year would be $ 637. 20

The amount that Benjamin's discounted premium would come to for the year is:

= 1, 080 x ( 1 - 66 %)

= $ 367. 20

The amount he would need to save every month on deployment is :

= 367. 20 / 9

= $ 40. 80

His total premium would be :

= 367. 20 + ( 1, 080 / 12 x 3 months when he comes back )

= $ 637. 20

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The probability that Sam is chosen first, Mike second, and Cindy third in a random order is 37!/40 (Option B).

The question is: Sam, Mike, and Cindy are in a lottery drawing for housing with 40 other students to choose their dorm rooms. If the students are chosen in random order, what is the probability that Sam is chosen first, Mike second, and Cindy third?

To find the probability, we need to consider the total possible ways the students can be chosen and the specific arrangement we want (Sam first, Mike second, and Cindy third). There are a total of 43 students, so there are 43! (43 factorial) ways to arrange them.

For the specific arrangement we want:
- There is 1 way to choose Sam first (out of 43 students).
- After choosing Sam, there is 1 way to choose Mike second (out of the remaining 42 students).
- After choosing Mike, there is 1 way to choose Cindy third (out of the remaining 41 students).

So, there is a total of 1 × 1 × 1 = 1 way to have the specific arrangement we want.

Now, we can calculate the probability by dividing the number of ways to get the specific arrangement by the total number of arrangements:

Probability = (1 way for the specific arrangement) / (43! total arrangements) = 1/(43!)

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The system of equations has one solution.

To determine whether the system of equations has one solution, no solution, or infinite solutions, we will compare the slopes and y-intercepts of the given equations:

Equation 1: [tex]y = (\frac{2}{3})-1[/tex]
Equation 2: y = -x + 4

Step 1: Identify the slopes and y-intercepts of each equation.
For Equation 1, the slope is 2/3, and the y-intercept is -1.
For Equation 2, the slope is -1, and the y-intercept is 4.

Step 2: Compare the slopes and y-intercepts.
The slopes are different (2/3 ≠ -1), and the y-intercepts are also different [tex](\frac{2}{3} ) ≠ 4[/tex].

Your answer: Since the slopes and y-intercepts are different, the system of equations has one solution.

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RSQ= 126 degrees

both angles are cooresponding angels therefore

5x+86=10x+46

86-46=10x-5x

40=5x

x=8

substitute

10(x)+46

10(8)+46

80+46

126

The total weight of the cards inside the box is approximately 217.6 oz.

Each deck of cards weighs 3.4 oz, and there are 64 decks of cards in the box. Therefore, the total weight of the cards inside the box is 3.4 oz/deck x 64 decks = 217.6 oz. As the answer needs to be rounded to the nearest ounce, we round 217.6 to the nearest ounce, which gives us 218 oz.

However, the question asks for the weight of the cards, which is only accurate to one decimal place. Therefore, we round 217.6 to one decimal place, which gives us 217.6 oz. Hence, the total weight of the cards inside the box is approximately 217.6 oz.

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We can decompose the given rational function as follows:

f(x) = (16x^3 + 12x^2 + 10x + 2) / [(x^4 – 4x^2)(x^2 + x + 1)^2(x^2 – 3x + 2)(x^4 + 3x^2 + 2)]

To find the partial fraction decomposition, we first factor the denominator completely:

x^4 – 4x^2 = x^2(x^2 – 4) = x^2(x – 2)(x + 2)

x^2 + x + 1 = (x + 1/2)^2 + 3/4

x^2 – 3x + 2 = (x – 1)(x – 2)

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

Substituting these factorizations into the denominator, we get:

f(x) = (16x^3 + 12x^2 + 10x + 2) / [x^2(x – 2)(x + 2)(x + 1/2)^2(3/4)^2(x – 1)(x – 2)(x^2 + 1)(x^2 + 2)]

We can now write the partial fraction decomposition as:

f(x) = A/x + Bx + C/(x – 2) + D/(x + 2) + E/(x + 1/2) + F/(x + 1/2)^2 + G/(x – 1) + H/(x^2 + 1) + I/(x^2 + 2)

where A, B, C, D, E, F, G, H, and I are constants to be determined.

Note that the term E/(x + 1/2) has a repeated linear factor (x + 1/2)^2, so we need to include a second term F/(x + 1/2)^2 in the decomposition.

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The equation of the tangent plane to the surface at the point (u,v) = (3,U) is z = x + 34 + U.

To calculate TT, we need to find the partial derivatives of O(u,v) with respect to u and v:

TT = (∂O/∂u) x (∂O/∂v)
  = (2, 0, 8) x (0, 0, 1)
  = (-8, 0, 0)

To find n(u,v), we normalize TT:

n(u,v) = TT/|TT|
      = (-1, 0, 0)

At the point u=3, v=U, O(u,v) = (2u + 0.0 - 40, 8u) = (-34, 24).

To find the equation of the tangent plane, we first find the normal vector to the plane, which is n(u,v) = (-1, 0, 0). Then we use the point-normal form of the equation of a plane:

(-1)(x + 34) + 0(y - 24) + 0(z - U) = 0
-x - 34 + z - U = 0
z = x + 34 + U

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Answer:

1/169

Step-by-step explanation:

The function has a local maximum at x = -6.

To determine the intervals on which the function f(x) = 4x³ - 12x² - 576x + 6 is increasing or decreasing, we first find its derivative, f'(x), and then analyze its critical points.

f'(x) = 12x² - 24x - 576

Now, set f'(x) = 0 and solve for x:

12x² - 24x - 576 = 0

Divide by 12:
x² - 2x - 48 = 0

Factor:
(x - 8)(x + 6) = 0

So, the critical points are x = 8 and x = -6.

Analyze the intervals:
f'(-7) > 0, so increasing on (-∞, -6)
f'(0) < 0, so decreasing on (-6, 8)
f'(9) > 0, so increasing on (8, ∞)

The function f(x) is decreasing on the interval (-6, 8). It is increasing on the interval (-∞, -6) and the interval (8, ∞). The function has a local maximum at x = -6.

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The number of days 3 3/4 bags of tortillas will last is 5 days.

To solve this problem, we need to use the concept of fractions. We know that Lucy's burrito stand uses 3/4 of a bag of tortillas every day. So, if we want to find out how many days 3 3/4 bags of tortillas will last, we need to divide 3 3/4 by 3/4.

To do this, we can convert 3 3/4 to an improper fraction, which is 15/4. Then, we can divide 15/4 by 3/4 using the following steps:

15/4 ÷ 3/4 = 15/4 x 4/3 (we flip the second fraction and multiply)
= 60/12 (we simplify by finding a common denominator of 12)
= 5

Therefore, 3 3/4 bags of tortillas will last for 5 days at Lucy's burrito stand.

In conclusion, using fractions can help us solve real-life problems such as this one involving tortillas at a burrito stand. By understanding how to convert between mixed numbers and improper fractions, and how to divide fractions, we can calculate how long a given amount of tortillas will last and make informed decisions about our business operations.

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To calculate ∑ (-1)^k pi^2k/2k, we can use the power series expansion for the cosine function:

cos(x) = ∑ (-1)^n x^(2n) / (2n)!

We can substitute pi^2 for x in this formula to get:

cos(pi^2) = ∑ (-1)^n (pi^2)^(2n) / (2n)!
= ∑ (-1)^n pi^(4n) / (2n)!

Now we can compare this to the original series we want to evaluate:

∑ (-1)^k pi^2k/2k = ∑ (-1)^n pi^(2n) / (2n)

We notice that the powers of pi in the two series match up, but the coefficients are different. However, we can use the identity cos(pi^2) = (-1)^n to rewrite the series we want to evaluate as:

∑ (-1)^n pi^(2n) / (2n) = ∑ (-1)^n pi^(4n) / (2n)! * (2n) / pi^2n

= pi^2 / 2 * ∑ (-1)^n pi^(4n) / (2n)!

Now we can substitute our expression for cos(pi^2) into this equation to get:

∑ (-1)^k pi^2k/2k = pi^2 / 2 * cos(pi^2)

= pi^2 / 2 * (-1)^n

Therefore, the value of the series is (-1)^n * pi^2 / 2.
To calculate the sum ∑ (-1)^k (pi^2k)/(2k), it is important to recognize that this is an alternating series with a general term given by a_k = (-1)^k (pi^2k)/(2k).

However, the question seems incomplete, as there is no specified range for the sum (i.e., the values of k). If you provide the range of k for which this sum is to be calculated, I would be glad to help you further.

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Therefore, the probability that Jack wins the race and Martina finishes last is 1/6 or approximately 0.167.

There are 3 people racing, so there are 3! = 6 possible ways the race can end (assuming no ties).

These are:

Jack, Martina, Napier

Jack, Napier, Martina

Martina, Jack, Napier

Martina, Napier, Jack

Napier, Jack, Martina

Napier, Martina, Jack

Of these 6 outcomes, there is only 1 where Jack wins the race and Martina finishes last: outcome 2.

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As per the given dimension, we need 343 small cubes to completely cover the larger cube.

To determine how many small cubes are needed to cover the larger cube, we need to think about how many of the smaller cubes can fit inside the larger cube.

We can start by looking at the dimensions of the larger cube. Each side is 7cm long, so the volume of the cube can be calculated by multiplying the length, width, and height:

7cm x 7cm x 7cm = 343 cubic centimeters

Now let's consider the dimensions of the smaller cubes. Each cube is 1cm x 1cm x 1cm, so the volume of each cube is:

1cm x 1cm x 1cm = 1 cubic centimeter

To determine how many of these smaller cubes are needed to cover the larger cube, we need to divide the volume of the larger cube by the volume of each small cube:

343 cubic centimeters ÷ 1 cubic centimeter = 343

So we need 343 small cubes to completely cover the larger cube.

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Complete Question:

How many cubes of dimensions 1cm*1cm*1cm are required to cover a cube of dimensions 7cm*7cm*7cm?

The zero of the function is 14. In the context of the situation, this means it will take 14 hours for Libby to completely drain the pool for cleaning.

To find the zero of the function, we need to determine the value of x when y equals 0.

0 = 10,080 - 720x

To solve for x, we will isolate the variable by following these steps:

1. Add 720x to both sides:
720x = 10,080

2. Divide both sides by 720:
x = 14

The zero of the function is 14, which means that after 14 hours, there will be no water remaining in the pool. In the context of the situation, this means it will take 14 hours for Libby to completely drain the pool for cleaning.

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If there is a one-to-one correspondence between two sets, we say that the sets have the same "size" or the same "cardinality."

Cardinality is a term used in mathematics and set theory to describe the size or number of elements in a set. It refers to the property of a set that determines how many objects it contains. For example, the cardinality of the set {1, 2, 3} is 3, because it contains three elements. The cardinality of a set can be finite (if it has a specific number of elements) or infinite (if it contains an uncountable number of elements).

Cardinality is usually denoted using the vertical bar notation |A|, where A is the set in question. For example, if A = {a, b, c}, then |A| = 3.

Cardinality is an important concept in mathematics, especially in areas such as set theory, combinatorics, and number theory. It is used to compare the sizes of different sets and to determine the properties of operations that involve sets, such as union, intersection, and complement.

If there is a one-to-one correspondence between two sets, we say that the sets have the same "size" or the same "cardinality."

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The complete table of values is

x   1    1.5     2     3    3.5     4     5

y  1.33 -1.58 -2.17 -1.33 -0.43 0.71 3.57

The equation of the function is given as

y = x²/3 + 6/x² - 5

To complete the table of values, we set x = 1, 1.5, 4 and 5

So, we have

y = 1²/3 + 6/1² - 5 = 1.33

y = 1.5²/3 + 6/(1.5²) - 5 = -1.58

y = 4²/3 + 6/(4²) - 5 = 0.71

y = 5²/3 + 6/(5²) - 5 = 3.57

The x and the y intervals are given as

0 ≤ x ≤ 5 and -5 ≤ y ≤ 4

See attachment for the graph and the labelled points

Estimating x²/3 + 6/x² - x - 3 = 0

We have

y = x²/3 + 6/x² - 5

Set y = x - 2

x²/3 + 6/x² - 5 = x - 2

So, we have

x²/3 + 6/x² - x - 3 = 0

This means that y = x - 2

From the graph, we have x = {1.28, 4.76}

Estimating x²/3 + 6/x² - x = 0

We have

y = x²/3 + 6/x² - 5

Set y = x - 5

x²/3 + 6/x² - 5 = x - 5

So, we have

x²/3 + 6/x² - x = 0

This means that y = x - 5

From the graph, we have x = undefined

It has no solution because the line does not intersect with the curve

Estimating x²/3 + 6/x² - 5 = 0

We have

y = x²/3 + 6/x² - 5

This means that y = 0

From the graph, we have x = {1.15, 3.69}

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Answer:

1

Step-by-step explanation:

The value of the rate of change when we put a glass of water at room temperature is 1/3.

The pace at which one quantity changes in relation to another quantity is known as the rate of change function. Simply said, the rate of change is calculated by dividing the amount of change in one thing by the equal amount of change in another.

The connection defining how one quantity changes in response to the change in another quantity is given by the rate of change formula. The formula for calculating the rate of change from y coordinates to x coordinates is y/x = (y2 - y1)/. (x2 - x1 ).

Rate of change  = change in temperature / time

= 10-5/15

=5 / 15

= 1/3

Therefore, the Rate of change is 1/3.

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Complete question;

What is the value of the rate of change when we put a glass of water at room temperature in the freezer for 15 minutes, what is its temperature at 5 minutes and then at 10 minutes.

The owners of the farm will stop harvesting when only 1000 trees remain during the fifth year of harvesting.

a) To get the regression equation for the relationship between time and trees remaining, we need to use linear regression. We can use the data given in the table to create a scatterplot and then find the line of best fit. Using a calculator or Excel, we can find that the regression equation is:
Trees remaining = 1177.38 - 36.25(time)
where "Trees remaining" is the number of trees remaining and "time" is the number of years since harvesting began.
b) To find during which year the owners of the farm will stop harvesting when only 1000 trees remain, we can substitute "1000" for "Trees remaining" in the regression equation and solve for "time":
1000 = 1177.38 - 36.25(time)
Solving for "time", we get:
time = (1177.38 - 1000) / 36.25
time ≈ 4.89 years
Therefore, the owners of the farm will stop harvesting when only 1000 trees remain during the fifth year of harvesting.

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The completed two column table in the question showing that the measure of the angle m∠GKB = 120° can be presented as follows;

Statement [tex]{}[/tex]                                                Reason

[tex]\overline{AB}[/tex] || [tex]\overline{EF}[/tex]         [tex]{}[/tex]                                          Given

m∠ELJ = 120°

m∠ELJ + m∠ELK = 180°     [tex]{}[/tex]                        Linear pair Postulate

m∠BKL + m∠GKB = 180°   [tex]{}[/tex]                        Linear pair Postulate

m∠ELJ + m∠ELK = mBKL + m∠GKB  [tex]{}[/tex]      Transitive property

∠ELK ≅ ∠BKL                         [tex]{}[/tex]                   1. Alternate Interior Angles

m∠ELK = m∠BKL [tex]{}[/tex]                                      2. Definition of congruent angles

m∠ELJ + m∠ELK = m∠ELK + m∠GKB[tex]{}[/tex]      Substitution property

m∠ELJ = m∠GKB[tex]{}[/tex]                                      Subtraction property

m∠GKB = m∠ELJ [tex]{}[/tex]                                     Symmetric property

m∠GKB = 120° [tex]{}[/tex]                                          Substitution

An angle is the figure formed at the point of intersection of two rays that have the same starting point. The parts of an angle includes; The vertex, which is the point of intersection of the rays, and the sides or arms of the angle, which are the two rays forming the angle.

The details of the the statements that completes the above table used to prove the measure of the angle m∠GKB = 120° are as follows;

Alternate interior angles theorem

The alternate interior angles theorem states that the alternate interior angle formed by the two parallel lines and their shared transversal are congruent.

Definition of congruent angles

Congruent angles are angles that have the same measure.

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1 marble is equal in weight to 84 jujubes.

Let's start by writing down the given information in equations:

5m = 3s (where m represents one marble and s represents one toy soldier)

7s = 5c (where c represents one plush chipmunk)

3c = 14j (where j represents one jujube)

We want to find out how many jujubes equal the weight of one marble, so we need to eliminate all the other variables except for j and m. We can do this by using substitution and algebraic manipulation.

First, we can solve the second equation for s in terms of c:

7s = 5c

s = (5/7)c

Then, we can substitute this expression for s in the first equation:

5m = 3s

5m = 3(5/7)c

m = (3/7)c

Next, we can solve the third equation for c in terms of j:

3c = 14j

c = (14/3)j

Now we can substitute this expression for c in the previous equation:

m = (3/7)c

m = (3/7)(14/3)j

m = 2j

So we have found that one marble is equal in weight to 2 jujubes. But the question asks for the weight of one marble in terms of jujubes, not in terms of jujubes and toy soldiers and plush chipmunks. We can use the other equations to eliminate the other variables:

5m = 3s

5m = 3(5/7)c

5m = (15/7)c

m = (3/7)c

7s = 5c

7s = 5(14/3)j

s = (10/3)j

Putting this all together:

m = (3/7)c

m = (3/7)(7s/5)

m = (3/5)s

m = (3/5)(10/3)j

m = 2j

So we have found that one marble is equal in weight to 2 jujubes. Finally, we can use the third equation to find how many jujubes are equal in weight to 1 marble:

3c = 14j

c = (14/3)j

m = (3/7)c

m = (3/7)(14/3)j

m = 2j

1 marble = 2 jujubes

1 jujube = 1/2 marble

1 marble = 2 jujubes = 2(84) = 168 jujubes

Therefore, one marble is equal in weight to 84 jujubes.

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The cost for one donut is $0.32, so the cost for one dozen donuts is:

12 donuts x $0.32/donut = $3.84

The cost for the cardboard box is $0.18 per square foot of cardboard, and there are 144 square inches in 1 square foot, so the cost per square inch of cardboard is:

$0.18 / 144 sq in = $0.00125/sq in

If t represents the total surface area of the box in square inches, then the cost of the box is:

t x $0.00125/sq in

To convert square inches to square feet, we divide by 144:

t/144 square feet x $0.18/square foot = t x $0.00125/sq in

Thus, the expression to model the total cost for one dozen donuts where t represents the total surface area of the box in square feet is:

$3.84 + (t/144) x $0.18

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There are 86,400 ways to arrange 5 different green balls and 7 different red balls in a row if all the green balls are separated.

If all the green balls are separated, we can think of the green balls as dividers that separate the red balls into groups. Since there are 5 green balls, there will be 6 groups of red balls. For example, if there are 7 red balls, the arrangement might look like this:

| R R R R R R R |

The "|" symbols represent the green balls. Each group of red balls is between two green balls.

To count the number of arrangements, we can think of each group of red balls as a box, and the green balls as dividers between the boxes.

We can arrange the 6 boxes in a row in 6! = 720 ways, and we can arrange the 5 green balls in the remaining 5 positions in 5! = 120 ways. Therefore, the number of arrangements is:

6! x 5! = 720 x 120 = 86,400

So ,there are 86,400 ways to arrange 5 different green balls and 7 different red balls in a row if all the green balls are separated.

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Answer: 90 feet squared

Step-by-step explanation:

The formula for finding the area of a triangle is

(bxh)/2

In this problem, you need to convert 3 yards to feet. The conversion is as follows:

1 yard = 3 feet

Therefore, 3 yards = 9 feet.

You can then plug everything in and solve as follows:

20x9= 180

Then divide 180 by 2.

180/2=90

Answer:[tex]90ft^{2}[/tex]

Step-by-step explanation:

Area of a triangle: A=(bh)/2

b = 20 ft

h = 3 yds = 9ft

Plug in

A=[(20ft)(9ft)]/2

A=180ft/2

A=90ft^2

Answer: Let's use the Pythagorean theorem to solve this problem.

Let x be the common factor of the ratio 3:4, so the sides of the rectangle are 3x and 4x.

The Pythagorean theorem states that for any right triangle, the sum of the squares of the two shorter sides is equal to the square of the length of the hypotenuse (the longest side).

So, for the rectangle with sides 3x and 4x, we have:

(3x)^2 + (4x)^2 = (diagonal)^2

9x^2 + 16x^2 = 100

25x^2 = 100

x^2 = 4

Taking the square root of both sides, we get:

x = 2

Therefore, the sides of the rectangle are:

3x = 3(2) = 6 cm

4x = 4(2) = 8 cm

So, the length and width of the rectangle are 6 cm and 8 cm, respectively.