16 less than one-fourth of some​ number, w" can be expressed algebraically as

Final answer:

The smallest possible dimension for the null space of a 13 by 91 matrix is 78. This is determined using the Rank-Nullity Theorem, taking into account that the rank of a matrix cannot exceed the number of its rows.

Explanation:

The question pertains to the dimension of the null space (also known as the nullity) of a matrix 'a.' The dimensions of matrix 'a' are 13 by 91, which means it has 13 rows and 91 columns. The null space of a matrix 'a' is the set of all vectors that, when multiplied by 'a,' give the zero vector. The dimension of the null space is referred to as the nullity of 'a.'

To find the smallest possible dimension of the null space, we consider the Rank-Nullity Theorem, which states that for any matrix 'A' of size m by n, the rank of 'A' plus the nullity of 'A' is equal to n, the number of columns in 'A.' The maximum rank a matrix can have is limited by the smaller of the number of rows or columns, so for matrix 'a' with dimensions 13 by 91, the maximum rank is 13 since there are only 13 rows.

Using the Rank-Nullity Theorem, we can say:

Therefore, the smallest possible dimension for the null space of matrix 'a' is 78.

Answer:

The company pays $60 a day for food and lodging and $0.65 for each mile traveled.

David drove 600 miles and was reimbursed $3,390.

Part A:

Let the number of days of trip be = x

Equation forms:

[tex]600(0.65)+60x=3390[/tex]

=> [tex]390+60x=3390[/tex]      .......(1)

Part B:

Solving (1) for x.

[tex]390+60x=3390[/tex]     (given)

Applying subtraction property of equality;

[tex]60x+390-390=3390-390[/tex]

Now simplifying this we get;

[tex]60x=3000[/tex]

Applying division property by dividing 60 on both sides, we get

x = 50

Part C:

We get x = 50, so David spent 50 days on the trip.

Since there is no angle restriction in this case, therefore the one rule that is applicable to this is that in forming a triangle, the sum of the lengths of the two smaller sides (A + B) should be larger than the length of the biggest side (C):

Triangle length rule: Side A + Side B > Side C

We can see that no matter how we combine the rods, the only combination of rods that satisfies this rule is:

 Triangle formed by rods 3 meters, 5 meters, and 7 meters

Therefore, there is only 1 triangle that can be formed from these four rods.

The best location for the satellite coordinates is

(Ax + Bx + Cx) / 3, (Ay + By + Cy) / 3).

We have,

From the figure given,

Assume that the coordinates of the three families are:

Sanchez = (a, b)

Perez = (c, d)

Reyes = (e, f)

The point equidistant from all three families' coordinates can be calculated using the formula.

Midpoint = ((m + o) / 2, (n + p) / 2)

Where (m, n) and (o, p) are the coordinates.

Midpoint between Sanchez and Perez:

Midpoint(SP) = A = ((a + c) / 2, (b + d) / 2)

Midpoint between Perez and Reyes:

Midpoint(PR) = B = = ((c + e) / 2, (d + f) / 2)

Midpoint between Sanchez and Reyes:

Midpoint(SR) = C = ((a + e) / 2, (b + f) / 2)

Equidistant Point

= (Ax + Bx + Cx) / 3, (Ay + By + Cy) / 3)

Where "Ax" represents the x-coordinate of the midpoint between Sanchez and Perez, Bx" represents the x-coordinate of the midpoint between Perez and Reyes, and Cx" represents the x-coordinate of the midpoint between Sanchez and Reyes. Similarly, Ay, By, and Cy" represent the y-coordinates of the respective midpoints.

Thus,

The best location for the satellite coordinates is

(Ax + Bx + Cx) / 3, (Ay + By + Cy) / 3).

Learn more about coordinates here:

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To solve this problem, let us assume linear motion so that we can use the equation:

t = d / v

where t is time, d is distance and v is velocity

First let us assign some variables, let us say that the velocity upstream is Vu while Velocity downstream is Vd, so that:

35 / Vu + 55 / Vd = 12                     ---> 1

30 / Vu + 44 / Vd = 10                     ---> 2

We rewrite equation 1 in terms of Vu:

(35 / Vu + 55 / Vd = 12) Vu

35 + 55 Vu / Vd = 12 Vu

12 Vu – 55 Vu / Vd = 35

Vu (12 – 55 / Vd) = 35

Vu = 35 / (12 – 55 / Vd)                  ---> 3

Also rewriting equation 2 to in terms of Vu:

Vu = 30 / (10 – 44 / Vd)                  ---> 4

Equating 3 and 4:

35 / (12 – 55 / Vd) = 30 / (10 – 44 / Vd)   

35 (10 – 44 / Vd) = 30 (12 – 55 / Vd)

Multiply both sides by Vd:

350 Vd – 1540 = 360 Vd – 1650

10 Vd = 110

Vd = 11 km / h

Using equation 3 to solve for Vu:

Vu = 35 / (12 – 55 / 11)

Vu = 5 km / h

Answers:

Vu = 5 km / h = velocity upstream

Vd = 11 km / h = velocity downstream

The expression 3(8 + 7) is equivalent to 45. The calculation follows the distributive property rule in mathematics, whereby we first simplify the expression inside the parentheses before multiplication.

The mathematical expression of 3(8 + 7) is based on the principle of distribution in mathematics. This principle can be interpreted as 'spread' or 'distribute' and applies when you multiply a number by addends within parentheses.

For the given expression 3(8 + 7), do the operation inside the parentheses first. So 8 + 7 equals 15. Now the expression becomes 3(15).

To find the solution, just multiply 3 by 15, which equals 45. So, 3(8 + 7) is equivalent to 45.

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The system of inequalities to determine the number of hamburger and hotdog buns Mr. Wu should purchase is x \\geq 2y for the quantity requirement, and 0.45x + 0.40y \\leq 60 for the budget constraint.

To determine how many hamburger buns (x) and hotdog buns (y) Mr. Wu can purchase for the concession stand, we need to set up a system of inequalities based on the given conditions. Since he needs at least twice as many hamburger buns as hotdog buns, we can express this requirement as an inequality: x \\geq 2y. Additionally, considering the cost of hamburger buns is $0.45 each and hotdog buns are $0.40 each, the total spending should not exceed $60. This gives us a budget constraint inequality: 0.45x + 0.40y \\leq 60. Therefore, the system of inequalities that can be used to determine the number of each kind of bun Mr. Wu should purchase is:

To solve this problem, we make use of the z statistic. We are to look for the bottom 8% who has the shortest lifespan, this is equivalent to a proportion of P = 0.08. Using the standard distribution tables for z, the value of z corresponding to this P value is:

z = -1.4

Now given the z and standard deviation s and the mean u, we can calculate for the number of years of the shortest lifespan:

x = z s + u

x = -1.4 (0.7) + 2.4

x = -0.98 + 2.4

x = 1.42 years

Therefore the life span is less than about 1.42 years

The plane was traveling at 319 miles per hour heading east. This was determined by setting up equations for the distances covered in both directions, considering the speed difference, and solving for the eastward speed.

To find the speed at which the plane was traveling east, we need to set up two equations based on the given information.

Let's denote the speed of the plane heading east  [tex]v_e[/tex] and the speed of the plane heading south [tex]v_s[/tex]. According to the problem, [tex]v_s[/tex] = [tex]v_e[/tex] + 45 mph. We also know that the plane flew south for 4 hours and east for 6 hours, covering a total distance of 3,370 miles.

To represent the sum of distances covered in both directions, we use the equation:

4[tex]v_s[/tex] + 6[tex]v_e[/tex] = 3,370

Substituting [tex]v_s[/tex] with [tex]v_e[/tex] + 45 in the equation yields:

4([tex]v_e[/tex] + 45) + 6[tex]v_e[/tex] = 3,370

By simplifying and solving for [tex]v_e[/tex], we find the speed of the plane traveling east. Let's solve it step by step:

4[tex]v_e[/tex] + 180 + 6[tex]v_e[/tex] = 3,370

10[tex]v_e[/tex] + 180 = 3,370

10[tex]v_e[/tex] = 3,190

[tex]v_e[/tex] = 319 mph

Therefore, the plane was traveling at 319 miles per hour heading east.

To calculate the number of different ways 30 qualifiers can be selected from 50 competitors in a ski jumping event, use the combination formula C(n, k) = n! / (k!(n - k)!), where in this case n=50 and k=30.

The question here is focused on finding the number of different combinations in which the qualifying round can be selected from a group of competitors in a sport event, specifically men’s ski jumping. This falls under the category known as combinatorics, which is a part of mathematics that deals with counting, both in a concrete and abstract way, as well as finding certain properties of finite structures.

The total number of different ways 30 competitors can be chosen from a group of 50 can be found using the combination formula, which is expressed as C(n, k) = n! / (k!(n - k)!), where "n" is the total number of competitors, "k" is the number of competitors to choose, "n!" signifies the factorial of "n", and "(n - k)!" is the factorial of the difference between "n" and "k".

In this situation, to find the number of different ways to select the 30 qualifiers from 50 competitors, we plug the values into the formula to calculate C(50, 30).

Final answer:

To find the number of different ways the qualifying round can be selected, you need to use combinations. The formula for combinations is C(n, r) = n! / (r!(n-r)!), where n is the total number of competitors and r is the number of competitors moving on to the qualifying round.

Explanation:

To find the number of different ways the qualifying round can be selected, we need to use combinations. The formula for combinations is C(n, r) = n! / (r!(n-r)!), where n is the total number of competitors and r is the number of competitors moving on to the qualifying round.

In this case, n = 50 and r = 30. Plugging these values into the formula, we get C(50, 30) = 50! / (30!(50-30)!). Simplifying this expression, we find that C(50, 30) = 211915132760.

Therefore, there are 211,915,132,760 different ways the qualifying round can be selected.

#include< stdio.h;
// #include< math.h;
int main() {
int I = 0;
double value[8];
for(I = 0; I < ;8; i++){
printf(“Enter value at index %d: “; (i +1 ));

Please note, always use %lf and not %f when reading double data from the user. double is %lf, while float is %f. That is the difference.

In this problem, we use two headers, stdio.h and math.h. The math.h defines a lot of mathematical functions. It returns double as a result since all functions in this library takes double as an argument.

The stdio.h header defines 3 variable types, namely size_t, FILE, and fpos_t.

The question asks for the required contact area between a suction cup and a ceiling to support an 80.0 kg person. Using principles of physics pressure calculations, a suction cup with a minimum contact area of 7.74 cm² would be needed.

The subject of your question is physics given it requires an understanding of pressure, force, and area relationships. To keep the suction cup adhered, the pressure difference between the lower (inside) of the suction cup and the outside (room pressure) must be large enough to support the weight of the person. This principle makes use of a simple physics equation: Pressure = Force/Area.

To support an 80.0-kg person, the force exerted due to weight would be mass multiplied by gravity or 80.0 kg * 9.8 m/s² = 784 N. The atmospheric pressure is about 101,325 Pascal (Pa) or N/m². Rearranging the equation for Pressure will give us the needed area: Area = Force/Pressure. So, the necessary contact area would be 784 N / 101325 Pa ≈ 0.00774 m² or 7.74 cm².

This means, that in ideal conditions and neglecting factors such as surface roughness, a suction cup with a contact area of at least 7.74 cm² would be needed to support an 80.0-kg person.

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Answer:

The number of identical containers are 27

Step-by-step explanation:

The picture below shows a container that Sue uses to freeze water.

We need to make 2000 cm³ of ice using small identical container.

Volume of cylinder [tex]=\pi r^2 h[/tex]

Where,

r = radius of cylinder (r=2 cm)

h = height of cylinder ( h=6 cm)

Volume of small cylinder [tex]=\pi (2)^2\cdot 6 = 75.39\text{ cm}^3[/tex]

We need to find number of small cylinder.

[tex]\text{Number pf small cylinder }=\dfrac{\text{Volume of ice}}{\text{Volume of each cylinder}}[/tex]

[tex]\text{Number pf small cylinder }=\dfrac{2000}{75.39}\approx 27[/tex]

Hence, The number of identical containers are 27

Answer:

65

Step-by-step explanation:

The solutions to the equation 15 tan^3 x=5 tan x in the interval [0, 2π) are approximately x = 0.6155, x = 3.757, x =2.527, x = 5.669.

The trigonometric equation provided in the question is 15 tan3 x=5 tan x. We can start solving this equation by dividing both sides by tan x, which gives 15 tan2 x = 5. Dividing again by 5, we get tan2 x = 1/3. The solutions to tan2 x = 1/3 are values of x in the interval [0, 2π) where the square of the tangent of x equals 1/3. However, these values cannot be easily computed, thus we use a calculator to approximate the results. We find the solutions to the equation by considering all angles whose tangent is either sqrt(1/3) or -sqrt(1/3). Therefore, the solutions for x in the interval [0, 2π) are approximately x = 0.6155, x = 3.757, x = 2.527, x = 5.669.

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to calculate circumference you can either use

 2 x PI x r

 or

pi x d

Answer:

PI X D

Step-by-step explanation:

Have a nice day :)