Dobrodošli na Primjer Web Stranice

\( \frac{ -73.50 }{4} \cdot \left(\frac{ -75 }{6} + \frac{7}{ 4 }\right) + 4.88 + 4.88 \)

Imena su : Lucas - Archie - Michael

Berechne \( 4 \times \left(\frac{1}{ 9 } + \frac{ 6 }{3}\right) \)

Ovo je primjer teksta koji kombinira običan tekst s LaTeX notacijom za matematičke formule. \( \quad f(x)=\displaystyle\log{\frac{x^2-3x+2}{x+1}} \) Na primjer, možemo prikazati sljedeću formulu:

Brzina svjetlosti u praznom prostoru definira se kao \(c = 299,792,458\) metara u sekundi.

Možemo također prikazati kvadratni korijen formule:

Kvadratni korijen iz broja \(x\) definira se kao \(\sqrt{x}\).

(1) The rectangular garden of Lucas is 45 m long and -73.50 m wide. Calculate the perimeter and area of the garden.

(2) Archie is cycling a distance of -75 km at an average speed of 4 km/h. How long will her ride take?

(3) A parabola has the equation \( f(x) = 4x^2 + 4.88x + 6 \). Find the coordinates of the vertex.

(4) In a coordinate system, the points A(9, -4) and B(7, 7) are given. Calculate the length of the line segment AB.

(5) The population of a city is growing exponentially at a rate of 1.01 annually. How many people will live in the city after 12 years, if the current population is 11000?

(6) Michael buys a laptop for 650 €. Each year, the laptop loses 25 % of its value. How much is the laptop worth after 4 years?

(7) A cone has a radius of 4.50 cm and a height of 6 cm. Calculate the volume of the cone.

(8) Sophia takes out a loan of 6500 € at an annual interest rate of 3 %. How much will the debt be after 2 years if no payments are made?

(9) The linear function \( f(x) = -5x + 5 \) intersects the x-axis at which point?

(10) The cube of Emma has an edge length of 10 cm. Calculate the volume and surface area of the cube.

(11) A figure is centrally dilated with center Z and dilation factor \( k = -1.50 \). Describe the effect of this dilation when \( k < 0 \) and when \( k > 0 \).

(12) The point A has coordinates \( A(2, 2) \). Calculate the image coordinates \( A' \) after a central dilation with center at the origin and factor \( k = 0.60 \).

(13) Arthur constructs a central dilation of the figure with center Z and dilation factor \( k = 5 \). The original figure has side lengths of 6 cm. How long is the corresponding side in the dilated figure?

(14) Two points B and B′ are centrally dilated with center Z(0,0). Point B has coordinates \( B(2, 4) \), and B′ has coordinates \( B'(6, -10) \). Calculate the dilation factor \( k \).

(15) A figure is dilated with a negative dilation factor \( k = -1 \). Explain the geometric meaning of this transformation regarding the orientation and position of the dilated figure.

(16) A figure has corners \( A(2, 8), B(3, -8), C(4, 1) \), and \( D(0, 8) \). Calculate the image coordinates of these points after a central dilation with center \( Z(0,0) \) and dilation factor \( k = 1.10 \).

(17) Emma has a rectangular area with side lengths \( a = 7 \) and \( b = 14. She dilates the rectangle centrally with center \( Z \) and dilation factor \( k = 0.90 \). Calculate the new side lengths and the area of the dilated rectangle.

(18) A right-angled triangular figure has the corners \( A(9, 9), B(10, -5) \) and \( C(-9, 4) \). Calculate the image of the triangle after a central dilation with center \( Z(0,0) \) and dilation factor \( k = 3 \). Also, determine the perimeter and area of the new triangle.

(19) A parabola has the equation \( f(x) = 2x^2 + 7x + 5 \). This parabola is centrally dilated with center \( Z(0,0) \) and dilation factor \( k = 0.80 \). Calculate the new coordinates of the vertex and the new behavior of the parabola.

(20) The points \( A(0, -14) \) and \( B(12, 11) \) are centrally dilated with the center at point \( Z(0, -1) \) and dilation factor \( k = 3.10 \). Calculate the dilation factor and the image coordinates of the points after dilation.

Term	Explanation
Center \( Z \)	The fixed point from which the dilation occurs. All image points are stretched relative to this point.
Dilation factor \( k \)	Indicates how much the figure is enlarged (\( k > 1 \)), reduced (\( 0 < k < 1 \)), reflected (\( k < 0 \)), or unchanged (\( k = 1 \)).
Behavior of the Parabola	The central dilation of a parabola changes its vertex and its width. If \( k > 1 \), the parabola becomes narrower; if \( 0 < k < 1 \), it becomes wider.
Area after Dilation	The area of the figure changes by the square of the dilation factor. If \( k \) is the dilation factor, the area changes by a factor of \( k^2 \).
Perimeter after Dilation	The perimeter of a figure changes linearly with the dilation factor \( k \). If the original perimeter is \( U_0 \), the new perimeter is \( U = k \cdot U_0 \).