By using two equations, 'Thomas's age = Huilan's age + 15' and 'Thomas's age + Huilan's age = 33', and substituting Huilan's age, we find that Thomas is 24 years old.
Explanation:
To solve this problem, we use a system of linear equations. Based on the given information, we can create two equations:
Thomas's age = Huilan's age + 15 (because we know Huilan is 15 years younger than Thomas)Thomas's age + Huilan's age = 33 (because we know the sum of their ages is 33)Replacing Huilan's age in the second equation with '(Thomas's age - 15)', we get: Thomas's age + (Thomas's age - 15) = 33
This simplifies to 2 * Thomas's age = 48, meaning Thomas's age = 24 years.
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Which expression is equivalent to 5a+20? A: 5(5a+4). B: 5(a+4). C: 5(a+20). D: 5(a+1)
Answer:
5(a+4)
Step-by-step explanation:
expression is equivalent to 5a+20
To get the equivalent expression we need to factor the given expression
5a+ 20
5a can be written as 5 * a
20 can be written as 5*2*2
WE can see that common factor is 5 for both 5a and 20
So GCF is 5
Now we factor out GCF 5
WE put 5 outside and write all the left of factors inside the parenthesis
5a +20
5 (a+2*2)
5(a+4)
Answer:
5(a+4)
Step-by-step explanation:
find the missing lenghts of the sides
Answer:
x = 8 ; y = 8√2
Step-by-step explanation:
In the given figure, two base angles are equal.
In an isosceles triangle, the sides opposite to the equal angles are equal.
∴ x= 8
The triangle is also right angled.
Using Pythagoras theorem,
hypotenuse² = base² + perpendicular²
y² = 8² + x² = 8² + 8² = 128
y = √128 = 8√2
∴ x = 8 ; y = 8√2
Answer:
option 2
Step-by-step explanation:
tan45° = x/8
=>1 = x/8
=>x =8
for y,
cos45° = 8/y
=>1/√2 = 8/y
=>y = 8√2
What polynomial should be subtracted from the polynomial y2–5y+1 to get the difference equal to: 0
please show work
Answer:
y = 5 or y = 0
Step-by-step explanation:
Solve for y over the real numbers:
y^2 - 5 y = 0
Factor y from the left hand side:
y (y - 5) = 0
Split into two equations:
y - 5 = 0 or y = 0
Add 5 to both sides:
Answer: y = 5 or y = 0
Answer:
y^2-5y+1
Step-by-step explanation:
If you have y^2-5y+1 and you need to subtract something from it to get 0, try it in parts. y^2-y^2=0, -5y+5y=0, 1-1=0. Remember that you are subtracting, so the y^2, the 5y, and the -1 you got are not the actual answers. Since the - in the parenthesis changes the signs, you need the remember to change the signs on the numbers you subtracted from the original numbers. So you get y^2. -5y, and 1. Put them together in a equation, and that's your answer.
what is the recursive formula for the geometric sequence with this explicit formula
an=9*(-1/3)^(n-1)
Answer:
a_0 = -27
a_n = a_(n-1) * (-1/3)
Step-by-step explanation:
First evaluate given formula at n=0 and specify that as starting value
Then find how to get from n-1 to n by comparing two values. In this case the next value is formed by multiplying by -1/3.
Answer:
[tex]a_n = a_{n-1} \cdot (-\frac{1}{3})[/tex]
Step-by-step explanation:
The explicit formula for the geometric sequence is given by:
[tex]a_n = a_1 \cdot r^{n-1}[/tex]
where,
[tex]a_1[/tex] is the first term
r is the common ratio to the following terms.
As per the statement:
Given the explicit formula for geometric sequence:
[tex]a_n = 9 \cdot (\frac{-1}{3})^{n-1}[/tex]
On comparing with [1] we have;
[tex]a_1 = 9[/tex] and [tex]r = -\frac{1}{3}[/tex]
The recursive formula for geometric sequence is given by:
[tex]a_n = a_{n-1} \cdot r[/tex]
Substitute the given values we have;
[tex]a_n = a_{n-1} \cdot (-\frac{1}{3})[/tex]
Therefore, the recursive formula for the geometric sequence is, [tex]a_n = a_{n-1} \cdot (-\frac{1}{3})[/tex]
1.) Of the 140 pages in a book, 25% of them have illustrations. How many pages have illustrations?
To determine how many pages have illustrations, 25% of the total 140 pages are calculated, resulting in 35 pages with illustrations.
Explanation:The student's question pertains to calculating the number of pages with illustrations based on a given percentage of the total pages in a book.
To find the answer, we need to calculate 25% of 140 pages. To do this, we multiply the total number of pages by the percentage of pages that have illustrations, remembering that 25% is the same as 0.25 in decimal form.
Therefore:
Number of pages with illustrations = 140 pages × 0.25 = 35 pages
So, there are 35 pages with illustrations in the book.
Use the Line Tool to graph the equation. 4x−6y=48 on a graph.
See attached.
Step-by-step explanation:When graphing equations presented in standard form, it is often convenient to convert them to intercept for. You do this by dividing by the constant on the right, and expressing the x- and y-coefficients as denominators:
... x/(x-intercept) + y/(y-intercept) = 1
Dividing your equation by 48, you get ...
... x/12 + y/(-8) = 1
That is, the intercepts of the line are (12, 0) and (0, -8). A line through these points will be the graph of the equation.
A technician is testing light bulbs to determine the number of defective bulbs. The technician records the table below to show the results. Result of Light Bulb Test Number of Bulbs Tested 14 28 84 336 Number of Defective Bulbs Found 1 2 6 ? The technician expects to find 24 defective bulbs when 336 are tested. Which statement explains whether the technician’s reasoning is correct, based on the information in the table? The reasoning is correct. The ratio of number of bulbs tested to defective bulbs is always 14 to 1. The reasoning is correct. The number of defective bulbs doubles, then triples, so the next number should be four times larger, regardless of the number of bulbs tested. The reasoning is not correct because the technician should have found the difference between 336 and 84, then divided the result by 6.
The reasoning is correct. The ratio of number of bulbs tested to defective bulbs is always 14 to 1.
Step-by-step explanation:We generally expect industrial processes to produce defects at about the same rate, meaning the proportion of defective product is generally considered to be a constant. Here, the proportion of defective bulbs is ...
... 1/14 = 2/28 = 6/84
so we expect it will be also 24/336. That is, the ratio of the number of bulbs tested to defective bulbs is expected to remain constant at about 14.
Answer:
A. The reasoning is correct. The ratio of number of bulbs tested to defective bulbs is always 14 to 1.
Step-by-step explanation:
The movie plex theater sold 1,456 tickets. 6 times as many regular tickets as discounted tickets? How many regular tickets were sold? All movies until 4:00 p.m require a discount ticket, and all movies after 4:00 p.m require a regular-price ticket.Each theater at the movieplex seats 98 people. What is the least number of showings the theater could have shown after 4:00 p.m last Saturday ? Explain how you found your answer
a) Out of each 7 tickets sold, 6 were regular tickets and 1 was a discount ticket. Thus the number of regular tickets sold was ...
... (6/7)×1456 = 1248 . . . regular tickets
b) At 98 seats per showing, it takes ...
... 1248 seats/(98 seats/showing) = 12.7347 showings
to accommodate all the ticket sold.
That is, the least number of showings there could have been is 13.
In the figure, what is the area of the shaded region?
Answer:
30 units ^2
Step-by-step explanation:
To find the area of the shaded region, we find find the area of the large triangle and subtract the area of the unshaded triangle.
A of large triangle = 1/2 b*h
height = (6+3) = 9
The base is found by using the pythagorean theorem c^2 = a^2 + b^2
We need to find b^2
c^2 -a^2 = b^2
taking the square root on each side
sqrt(c^2 -a^2) = sqrt(b^2)
the base = sqrt(c^2 -a^2)
= sqrt( 15^2 - 9^2)
= sqrt(225-81)
= sqrt(144)
=12
Now that we know the base and the height, we can find the area
A of large triangle = 1/2 b*h
= 1/2 * 12 * 9
= 6*9 = 54
Using the rule of similar triangles
9 6
---- = ----------
12 base
We can use cross products to find the base of the smaller triangle
9* base = 12*6
9* base = 72
Divide by 9 on each side
base = 72/9 = 8
Now we can find the area of the smaller triangle
base = 8 and height = 6
A of smaller triangle = 1/2 b*h
= 1/2 *8 * 6
= 4*6 = 24
Area of the shaded region = Area large triangle - Area of small triangle
= 54-24
= 30
If the perimeter of the rectangle is 30 cm and value of j is 8.
a) Create an equation that you can use to find the value of h
b) Solve the equation. What is the value of h?
PART A
Equation:__________________________
PART B
Solve the equation. Show your work
Answer: h =______________________
the amount y (in bales) of hay remaining after feeding cows for x days is y=-3.5x+ 105
The given linear equation describes how the amount of hay decreases by 3.5 bales for each day cows are being fed. The initial amount of hay is given as 105 bales.
Explanation:The equation provided, y=-3.5x + 105, is a linear equation wherein y represents the amount of hay remaining (in bales) and x represents the days of feeding cows. This equation tells us that for every day that passes (an increase in x by 1), the amount of hay decreases by 3.5 bales. When x = 0, which represents the beginning before any hay has been eaten, there are 105 bales of hay available.
To further illustrate, after one day (x = 1), the amount of hay left would be calculated as y=-3.5(1) + 105 = 101.5 bales. After two days (x = 2), y=-3.5(2) + 105 = 98 bales, and so on.
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someone help me pls.....
Answer:
[tex]\dfrac{-x+5}{6x^2-x-12}[/tex]
Step-by-step explanation:
The denominators are the same. You can add the numerators without any extra work.
[tex]=\dfrac{(4x+5)-(5x)}{6x^2-x-12}=\dfrac{-x+5}{6x^2-x-12}[/tex]
The denominator factors as (2x-3)(3x+4), so there are no factors that will cancel with the numerator.
HURRRYYYY
A. It is stretched horizontally by a factor of 2 and translated up 3.
B. It is compressed horizontally by a factor of 2 and translated up 3.
C. It is stretched vertically by a factor of 2 and translated up 3.
D. It is compressed vertically by a factor of 2 and translated up 3.
The first graph, X is multiplied by 2 (2x) which compresses the graph horizontally by the factor of 2.
Then 3 is added , which shifts the graph up 3 units.
The answer would be: B. It is compressed horizontally by a factor of 2 and translated up 3.
I cannot solve this. I don't know how.
The notation f(x) means you have a function that has been given the name f, and it makes use of the variable x. The variable in the parentheses is called the "argument" of the function f.
(a) To find f(q), you put q everywhere x is in the function equation. This is called evaluating the function for an argument of "q". In the following, note that we have simply changed x to q. (It's really that simple.)
... f(q) = q² -2q +3
(b) As in the previous case, we replace x with (x+h) everywhere.
... f(x+h) = (x+h)² -2(x+h) +3
You can multiply it out, but there appears to be no need to do so for this part of the question.
(c) The intent here is that f(x+h) and f(x) will be replaced by their values and the whole thing simplified. This requires you expand the expression you see in part (b), subtract f(x), collect terms, and divide the whole thing by h. You have to make use of what you know about multiplying binomials.
We can do it in parts:
... f(x+h) = (x+h)² -2(x+h) +3
... = (x² +2xh +h²) + (-2x -2h) +3
Separating the h terms, this looks like ...
... = (x² -2x +3) + (2xh -2h +h²)
Now, we can finish the numerator part of the expression by subtracting f(x):
... f(x+h) -f(x) = (x² -2x +3) +(2xh -2h +h²) -(x² -2x +3)
You can see that the stuff in the first parentheses matches that in the last parentheses, so when we subtract the latter from the former, we get zero. We are left with only the terms containing h.
... f(x+h) -f(x) = 2xh -2h +h²
To finish up this problem, we need to divide this numerator value by the denominator h.
... (f(x+h) -f(x))/h = (2xh -2h +h²)/h
... = (2xh)/h -(2h)/h +h²/h
... = 2x -2 +h . . . . . this is the value of the expression
... (f(x+h) -f(x))/h = 2x -2 +h
find the value of x. please help.
Answer:
20.4
Step-by-step explanation:
The rule for secants is pretty simple. The product of the distance to one intersection with the circle and the distance to the other intersection with the circle is a constant. (This same rule applies when the secants intersect outside the circle.)
Here, that means ...
8 × 23 = 9 × x
x = 8·23/9 ≈ 20.4 . . . . . divide by 9
Will someone please help me with the 3 highlighted questions?
6. f(1)+g(2) = 4
8. g(4)-f(0) = 18
10. 2g(-4) = 22
Step-by-step explanation:Put the number where the variable is and do the arithmetic.
6.f(1) = 2·1 -5 = -3
g(2) = |-3·2-1| = |-7| = 7
f(1) + g(2) = -3 + 7 = 4
___
8.g(4) = |-3·4 -1| = |-13| = 13
f(0) = 2·0 -5 = -5
g(4) - f(0) = 13 -(-5) = 18
___
10.g(-4) = |-3·(-4)-1| = |11| = 11
2g(-4) = 2·11 = 22
The shortest side of an isosceles triangle is 4x−2 inches long. The two longer sides are 5 inches longer than the shortest side. The perimeter of the triangle is 64 inches.
What is the length of the longer sides of the triangle?
Answer:
23 inches
Step-by-step explanation:
If we add 5 inches to the shortest side, all sides will be the same length and the perimeter will be 69 inches. The longest sides have length that is 1/3 that, or ...
... (1/3)·(64 +5 in) = 23 in
_____
You can solve for x, but you obviously don't need to.
The perimeter is ...
... (4x -2) + 2(4x -2 +5) = 64
... 12x +4 = 64
... 12x = 60
... x = 5
... 4x -2 +5 = 4·5 +3 = 23 . . . . the length of the longest side in inches
find all the zeros in the equation x^4-6x^2-7x-6=0
show your work
To find the zeros of the polynomial equation, the equation has to be transformed into a quadratic equation. Using the quadratic formula, the roots of the quadratic can be found. Finally, substituting these roots again into the original given equation will give the solutions of the polynomial.
Explanation:To find the zeros of the equation x^4-6x^2-7x-6=0, we must use factoring or the quadratic formula. The quadratic formula is -b ± √b² - 4ac 2a, where a, b, and c are coefficients of the equation. However, since this equation is a quartic and not a quadratic, we first need to simplify it into a quadratic form, which can be more easily solved.
Let's write the equation as (x^2)^2 - 6*(x^2) - 7x - 6 = 0 and let y = x^2. So the equation becomes y^2 - 6y - 7x - 6 = 0. Solve this equation as a quadratic. After finding the values of y, substitute y = x^2 back into them to solve for x.
Without the original quadratic equation or an exact simplified equation, we are not able to provide concrete solutions. Applying the mentioned steps will help you find the precise solutions.
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How many solutions does the system have?
y=−2x−4
y=3x+3
A. One solution
B.No solutions
C. Indefinitely about of solutions
Answer: one solution
Step-by-step explanation:
The two lines have different slopes, so they can't be the same line (infinitely many solutions) or parallel (no solutions). The lines intersect at
3x+3 = -2x - 4
5x = -7
x = -7/5
3(-7/5) + 3 = -2(-7/5) - 4
-21 + 15 = 14 - 20
-6 = -6
Answer:
The answer is one solution
Step-by-step explanation:
I tried it on khan and it was correct.
someone help pls need help on this one
Answer:
[tex]\dfrac{2}{3x^5y}[/tex]
Step-by-step explanation:
A negative exponent in the numerator is the same as a positive exponent in the denominator, and vice versa.
... a^-b = 1/a^b . . . . . for any value of b, positive or negative
The exponent of a product is the sum of the exponents:
... (a^b)(a^c) = a^(b+c)
___
Applying these rules, you have
... = 2/(3x^4·x·y) = 2/(3x^(4+1)·y) = 2/(3x^5·y)
graph the function. please help asap
Answer:
See the attached
Step-by-step explanation:
A graph of almost any exponential function quickly goes off-scale. The attachment shows a short table of values.
what are the real and complex solutions of the polynomial equation? x^4-41x^2=-400
show your work please
The real solutions are -5, -4, 4, 5. There are no complex solutions.
Step-by-step explanation:The equation ...
... x^4 -41x^2 +400 = 0
can be factored as ...
... (x^2 -16)(x^2 -25) = 0
... (x -4)(x +4)(x -5)(x +5) = 0
So, all roots are real and are ...
... x ∈ {-5, -4, 4, 5}
_____
These are the values of x that make the factors zero.
Final answer:
The real solutions of the polynomial equation x^4 - 41x^2 = -400 are x = 4, -4, 5, and -5. There are no complex solutions since the equation can be factored into real numbers without the need for complex terms.
Explanation:
To find the real and complex solutions of the polynomial equation x^4 - 41x^2 = -400, we can begin by rewriting the equation in a more familiar quadratic form. Adding 400 to both sides gives us:
x^4 - 41x^2 + 400 = 0
We can set y = x^2, which turns our equation into:
y^2 - 41y + 400 = 0
Factoring this quadratic equation, we get:
(y - 16)(y - 25) = 0
So, y = 16 or y = 25. Since y = x^2, we solve for x:
x^2 = 16 → x = ±4
x^2 = 25 → x = ±5
Therefore, the real solutions are x = 4, -4, 5, -5. There are no complex solutions in this case because all values of x are real numbers.
A boat traveled 210 miles downstream and back. The trip downstream took 10 hours. The trip back took 70 hours. What is the speed of the boat in still water? What is the speed of the current?
The speed of the boat in still water is 12 mph, and the speed of the current is 9 mph.
Let's denote the speed of the boat in still water as ( b ) and the speed of the current as ( c ).
Downstream Trip:
The speed of the boat relative to the water is ( b + c ), and the distance is 210 miles.
Therefore, the time taken downstream is:
[tex]\[ \text{Time} = \frac{\text{Distance}}{\text{Speed}} = \frac{210}{b + c} \][/tex]
Given that this trip took 10 hours, we have:
[tex]\[ 10 = \frac{210}{b + c} \][/tex]
Upstream Trip:
The speed of the boat relative to the water is ( b - c ), and the distance is again 210 miles.
Therefore, the time taken upstream is:
[tex]\[ \text{Time} = \frac{\text{Distance}}{\text{Speed}} = \frac{210}{b - c} \][/tex]
Given that this trip took 70 hours, we have:
[tex]\[ 70 = \frac{210}{b - c} \][/tex]
Now, we have a system of two equations with two variables:
[tex]\[ 10 = \frac{210}{b + c} \][/tex]
[tex]\[ 70 = \frac{210}{b - c} \][/tex]
Solving this system of equations will give us the values of ( b ) and ( c ), which represent the speed of the boat in still water and the speed of the current, respectively. Let's solve it:
From the first equation:
[tex]\[ b + c = \frac{210}{10} = 21 \][/tex]
From the second equation:
[tex]\[ b - c = \frac{210}{70} = 3 \][/tex]
Adding the two equations:
(b + c) + (b - c) = 21 + 3
2b = 24
b = 12
Substituting ( b = 12 ) into ( b + c = 21 ):
12 + c = 21
c = 21 - 12
c = 9
So, the speed of the boat in still water is 12 mph, and the speed of the current is 9 mph.
create two similar, but not equal, right triangles using A (-5,-1) and b(4,3.5)
Answer:
The attachment shows ΔBAC ~ ΔBDA
Step-by-step explanation:
You want segment AB to be part of two similar, but not congruent, triangles. One way to do that is to make AB the hypotenuse of one triangle and the leg of another.
It is convenient to construct these triangles using point M as the arbitrary midpoint of the hypotenuse of the larger triangle. (We don't know the coordinates of M—we just know it is on the perpendicular bisector of AB.) BC is a diameter of circle M, and AD is the altitude of ΔABC.
please help just looking for the answer
For this case, we have that by definition:
Let "x" be an angle of any vertex of a right triangle.
[tex]Sin (x) = \frac {Cathet \ opposite} {hypotenuse}[/tex]
So, if we want to find the sine of angle A:
[tex]Sin (A) = \frac {3} {5}[/tex]
Thus, the sine of angle "A" is[tex]\frac {3} {5}[/tex]
Answer:
[tex]\frac {3} {5}[/tex]
Option A
Solve the proportions using cross products. Round to the nearest hundredth is necessary. 21miles/49hours = 15miles h hours
[tex]\dfrac{21}{49}=\dfrac{21:7}{49:7}=\dfrac{3}{7}\\\\\dfrac{21}{49}=\dfrac{15}{h}\to\dfrac{3}{7}=\dfrac{15}{h}\qquad\text{cross multiply}\\\\3h=(7)(15)\\\\3h=105\qquad\text{divide both sides by 3}\\\\\boxed{h=35}\\\\Answer:\ 35\ hours[/tex]
The coordinates of the vertices of a triangle are (1, 8) , (9, 8) , and (9, −2) . What are the coordinates of the circumcenter of the triangle? Enter your answer in the boxes.
Answer:
(5, 3)
Step-by-step explanation:
The given coordinates define a right triangle with (9, 8) as the vertex where the right angle is located. Then the other two coordinates define a diameter of the circumcircle. Its midpoint is the center.
... center = ((1, 8) +(9, -2))/2 = (5, 3)
A square has a length of 3/4m. Find the area.
Answer:
9/16
Step-by-step explanation:
Since the area of a square is s^2, then ther area of this square is (3/4)^2 is 9/16.
Find the measurement of one interior angle in each polygon. Round your answers to the nearest tenth if necessary.
Find the measure of one exterior angle in each polygon. Round your answer to the nearest tenth if necessary.
1–4: The interior angle of an n-sided regular polygon is (n-2)/n times 180°. For n ∈ {5, 6, 11, 4}, this is {3/5, 4/6, 9/11, 2/4} · 180°, or {108°, 120°, 147.3°, 90°}
5–8: The exterior angle of an n-sided regular polygon is 360°/n. For n ∈ {5, 9, 7, 4}, this is 360°/{5, 9, 7, 4}, or {72°, 40°, 51.4°, 90°}
Please help if you can
Answer:
rational: √81, √121
irrational: √89, √131
Step-by-step explanation:
You know your squares, so you know that 81 = 9² is a perfect square. Its square root is 9, a rational number.
And you know that 121 = 11², another perfect square. Its square root is 11, a rational number.
The remaining numbers are not the roots of squares of integers, so will be irrational.